Versio* *n 1 23 December 20* *02 The stable moduli space of Riemann surfaces: Mumford's conjecture Ib Madsen Michael Weiss Institute for the Mathematical Sciences Department of Mathematics Aarhus University Aberdeen University 8000 Aarhus C, Denmark Aberdeen AB24 3UE, United Kingdom email: imadsen@imf.au.dk email: m.weiss@maths.abdn.ac.uk 1 Contents 1 Introduction: Results and methods 4 1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . 4 1.2 A geometric formulation . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . 5 1.3 Outline of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . 8 2 Some generalized bundle theories 12 2.1 The basic sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . 12 2.2 Homotopy theory of sheaves . . . . . . . . . . . . . . . . . . . . . .* * . . . . . 16 2.3 Different models and monoid structures . . . . . . . . . . . . . . . .* * . . . . 18 3 The spaces of diagram (2.3) 20 3.1 A cofiber sequence of Thom spectra . . . . . . . . . . . . . . . . . * *. . . . . 20 3.2 The spaces |hW| and |hV| . . . . . . . . . . . . . . . . . . . . . . * *. . . . . 24 3.3 The space |hWloc| . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . 29 3.4 The space |Wloc| . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . 32 4 Application of Vassiliev's h-principle * * 34 4.1 Sheaves with category structure . . . . . . . . . . . . . . . . . . . * *. . . . . . 34 4.2 Armlets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . 37 4.3 Proof of theorem 1.3.1 . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . 40 5 Some homotopy colimit decompositions 43 5.1 Description of main results . . . . . . . . . . . . . . . . . . . . .* * . . . . . . 43 5.2 Morse singularities, Hessians and surgeries . . . . . . . . . . . . . * *. . . . . . 45 5.3 Second row . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . 48 5.4 Third row . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . 52 5.5 Fourth row, right hand column . . . . . . . . . . . . . . . . . . . . * *. . . . . 55 5.6 Fourth row, left hand column . . . . . . . . . . . . . . . . . . . . .* * . . . . . 56 5.7 Using the concordance lifting property . . . . . . . . . . . . . . . .* * . . . . . 60 2 6 The connectivity problem 61 6.1 Overview and definitions . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . 61 6.2 Categories of multiple surgeries . . . . . . . . . . . . . . . . . . .* * . . . . . . 63 6.3 Parametrized multiple surgeries . . . . . . . . . . . . . . . . . . . * *. . . . . . 65 6.4 Annihiliation of 2-spheres . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . 68 7 Stabilization * * 70 7.1 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . 70 7.2 Using the Harer-Ivanov stability theorem . . . . . . . . . . . . . . * *. . . . . 72 A More about sheaves 73 A.1 Concordance and the representing space . . . . . . . . . . . . . . . .* * . . . . 73 A.2 Categorical properties . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . 75 A.3 Relative homotopy and fibrations . . . . . . . . . . . . . . . . . . .* * . . . . . 76 B Sheaves with a category structure 77 B.1 Cocycle sheaves without indices . . . . . . . . . . . . . . . . . . . * *. . . . . . 77 B.2 A variation on the nerve construction . . . . . . . . . . . . . . . .* * . . . . . 80 B.3 Completion of the proof . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . 80 C Geometric realizations and the bar construction 83 C.1 Realization, quasifibrations and homology fibrations . . . . . . . . .* * . . . . 83 C.2 The bar construction for monoids without unit . . . . . . . . . . . . * *. . . . 85 D Generalities about homotopy colimits and stratifications 86 D.1 Homotopy colimits . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . 86 D.2 Stratifications and homotopy colimit decompositions . . . . . . . . . * *. . . . 88 3 1 Introduction: Results and methods 1.1 Results The main result of this paper amounts to a complete evaluation of the integral * *cohomological structure of the stable mapping class group. In particular it verifies the con* *jecture of D. Mumford about the rational cohomology of the mapping class group: H*(B g,b; Q) = Q[~1, ~2, . .]. for2* < g - 1 where g,bis the mapping class group of an oriented surface Fg,bof genus g with* * b boundary circles (and no punctures). The ~i are the Miller-Morita-Mumford ät utological* *" classes of degree 2i. For b > 0, the standard homomorphisms g,b ! g+1,b, g,b ! g,b-1 (1* *.1) yield maps of classifying spaces that induce isomorphisms in integral cohomolog* *y in degrees less than g=2 - 1 by the stability theorems of Harer [17] and Ivanov [21]. The * *colimit of the maps B g,b-! B g+1,b-! B g+2,b-! . . . will be denoted B 1,b, B 1,b = colimB g,b' hocolim B g,b. g g The groups g,bare perfect for g > 1, so B 1,b has a perfect fundamental group * *and one may apply Quillen's plus construction to it. The result is independent of b up * *to homotopy equivalence, so we denote it by B 1+. A celebrated result from [40] asserts tha* *t Z x B 1+ and B 1+ are infinite loop spaces, so that homotopy classes of maps to either o* *f these spaces form the degree 0 part of a generalized cohomology theory. Next we review a completely different infinite loop space, one which is rather * *well known to homotopy theorists. Let us write Gr 2(R2+n) for the Grassmann manifold of or* *iented 2- dimensional subspaces of R2+n . There are two canonical bundles over Gr2(R2+n),* * namely, the tautological 2-plane bundle Ln and its n-dimensional orthogonal complement * *L?n. The restriction fi L?n+1fiGr2(R2+n) is the direct sum of L?n and a trivialized real line bundle. This yields an in* *clusion of associated Thom spaces, S1 ^ Th (L?n) -! Th (L?n+1) 4 and hence a sequence of maps (in fact cofibrations) . .-.! n+1Th (L?n-1) -! n+2Th (L?n) -! n+3Th (L?n+1) -! . . . whose colimit is traditionally denoted 1 CP1-1= colim n+2Th (L?n). n There is a map ff1 : Z x B 1+- ! 1 CP1-1 constructed and examined in considerable detail in [25]. Our main result is the* * following theorem conjectured in [25]: Theorem 1.1.1 The map ff1 : Z x B 1+- ! 1 CP1-1 is a homotopy equivalence. The cohomological structure of 1 CP1-1is completely known both with Q coeffici* *ents and with Fp coefficients for all p, so the theorem gives the cohomology of B 1+ and* * hence of B 1,b with these coefficients. The space 1 CP1-1fits into the homotopy fibration sequence of [33], 1 CP1-1 - !--! 1 S1 (CP+1) - -@-! 1+1 S1 (1* *.2) where the subscript + denotes an added disjoint base point. The structure of th* *e cohomo- logy H*( 1 CP1-1; Fp) was recently determined in [11]. It is rather involved an* *d we refrain from listing the result. The rational structure is much easier to describe. The homotopy groups of 1+1 S1 are equal to the stable homotopy groups of sphe* *res, up to a shift of one, and are therefore finite. Thus H*(!; Q) is an isomorphism. T* *he canonical complex line bundle over CP 1, considered as a map from CP 1 to {1} x BU induce* *s via Bott periodicity a map L: 1 S1 (CP+1) -! Z x BU, and L is a rational equivalence. It follows that the rational cohomology of a c* *omponent, say 10CP-11, is equal to the rational cohomology of BU, and hence by theorem 1* *.1.1 that H*(B 1,b; Q) ~=H*(BU; Q). This yields Mumford's conjecture. 1.2 A geometric formulation Fix an integer d 0. Let ß :M ! X be a smooth fiber bundle with oriented d-dim* *ensional fibers. We assume that the fibers are closed. There are two canonical vector bu* *ndles on M , namely the vertical tangent bundle T iM and the stable vertical normal bundle N* *i M . The 5 latter is defined to be the normal bundle of a fiberwise embedding of M into X * *x Rd+n for large n. Let Grd(Rd+n) be the Grassmann manifold of oriented d-dimensional subs* *paces of Rd+n , and let Ud,nand U?d,nbe the two standard vector bundles over it of dimen* *sion d and n, respectively. The vertical tangent bundle and the vertical normal bundle are* * classified by bundle maps T iM -! Ud,n, Ni M -! U?d,n, respectively, for large n. We can view Ni M as a tubular neighborhood of M in X* * x Rd+n and obtain the fiberwise Thom-Pontryagin map X+ ^ Sd+n ~=Th (X x Rd+n) -! Th (Ni M) -! Th (U?d,n), by mapping the complement of Ni M in Th (X x Rd+n) to 1 2 Th (Ni M). Generalizi* *ng the description of 1 CP1-1given earlier, but switching to different notation, * *we put 1 hV = colimn d+nTh (U?d,n). The adjoint of the above composition`gives a homotopy class of maps from X to * *1 hV. The universal case is X = BDiff(F ), where F runs over over a set of represe* *ntatives of the diffeomorphism classes of closed, smooth and oriented d-manifolds. In th* *is case we obtain a ff: BDiff(F ) -! 1 hV . (1* *.3) In the case d = 2, it is convenient to make extra assumptions. For example, we* * may wish to consider only connected oriented surfaces F with an embedded copy of S0* * x D2, and`diffeomorphisms F ! F relative to the embedded S0 x D2. With these conventi* *ons, BDiff(F ) becomes an A1 -monoid under`connected sum, and the map ff can be s* *hown to factor over the group completion of BDiff(F ). Each Diff(F ) has contract* *ible com- ponents by [9], [10]. The group of components`is the mapping class group g,2wh* *ere`g is the genus of F . Hence in this case BDiff(F ) becomes homotopy equivalent to * * gB g,2. The group completion is a + B B g,2 ' B 1 x Z , (1* *.4) g cf. [25], [40], and ff of (1.3) induces the map ff1 of theorem 1.1.1. Let us return to the general case, with fixed d 2 N. We give a geometric interp* *retation of homotopy classes of maps from a smooth manifold X without boundary into 1 hV. * *We represent such a homotopy class by a pointed map X+ ^ Sd+n -! Th (U?d,n) for some large n, transverse to the zero section of U?d,n. The resulting inver* *se image of the zero section is a submanifold M X x Rd+n , of dimension dim (X) + d, with* * a map 6 vM : M ! Gr d(Rd+n). The projection ßM : M ! X is proper, since M is close* *d in Th (X x Rd+n). The normal bundle of M in X x Rd+n is identified with v*MU?d,n,* * so T M x Rd+n ~= T M v*MU?d,n v*MUd,n~=(ß*MT X x Rd+n) v*MUd,n. (1* *.5) Standard obstruction theory now implies that T M x R ~= (ß*MT X x R) v*MUd,n. (1* *.6) We set E = M xR, write ßE for the composition E ! M ! X and vE for the composit* *ion E ! M ! Grd(Rd+n), and obtain from (1.6) a surjective bundle map ^ßE: T E ! ß*ET X . Since E is open, the submersion theorem of Phillips [30], [16], [15] applies, s* *howing that the pair (ßE , ^ßE) is homotopic through vector bundle surjections to a pair co* *nsisting of a submersion ß :E ! X and its differential dß :T E ! ß*T X . For us it is also im* *portant to ensure that the underlying homotopy E x [0, 1] ! X combines with the projec* *tion f :E ! R to give a proper map E x [0, 1] ! X x R. This is trivially the case wh* *en X is closed, in particular when X is a sphere, because then the projection f :E ! R * *is proper. In the general case a more careful application of the submersion theorem is req* *uired; we omit the details. There is an additional feature in this situation. Namely, the vertical tangent * *bundle of the submersion ß :E ! X is identified with v*EUd,nx R and therefore projects to a t* *rivial line bundle. In terms of the vertical 1-jet bundle p1i:J1i(E, R) -! E whose fiber at z 2 E consists of all affine maps from the vertical tangent spac* *e (T iE)z to R, this feature together with f :E ! R amounts to a section f^ of p1isuch th* *at f^(z): (T iE)z ! R is surjective for every z 2 E . We introduce the notation hV(X) for the set of pairs (ß, ^f), where ß :E ! X is* * a smooth submersion with (d + 1)-dimensional oriented fibers and f^:E ! J1i(E, R) is a s* *ection of p1iwith underlying map f :E ! R, subject to two conditions: for each z 2 E the * *affine map f^(z): (T iE)z ! R is surjective, and (ß, f): E ! X x R is proper. Concordance defines an equivalence relation on hV(X). Let hV[X] be the set of e* *quivalence classes. Then we have a natural bijection hV[X] ~=[X, 1 hV] . (1* *.7) There is a similar but easier interpretation of homotopy classes of maps from X* * to the source of (1.3). Namely, let V(X) be the set of pairs (ß, f) with ß as before and f :* *E ! R a smooth function, subject to two conditions: the restriction of f to any fib* *er of the submersion ß :E ! X is regular (= nonsingular), and (ß, f): E ! X x R is proper* *. Let V[X] be the set of concordance classes of elements in V(X). Then a V[X] ~=[X, BDiff(F ) ] (1* *.8) 7 ` with BDiff(F ) as in (1.3). Indeed, an element of V(X) is a proper submersi* *on with target X x R, hence a smooth fiber bundle on X x R by Ehresmann's fibration the* *orem. An element (ß, f) 2 V(X), with ß :E ! X , determines a section j1if of the proj* *ection J1i(E, R) ! E by fiberwise 1-jet prolongation. The map V(X) -! hV(X) ; (ß, f)7! (ß, j1if) (1* *.9) respects the concordance relation and so induces a map V[X] ! hV[X], which corr* *esponds to ff in (1.3) under the isomorphisms (1.7) and 1.8). When d = 2, it is a good* * idea to modify the source of (1.9); we will return to this point in a little while. The new description of ff reformulates theorem 1.1.1 as a statement about integ* *rability of certain jet bundle sections, up to homotopy or concordance. Statements of this* * type are called h-principles [14]. 1.3 Outline of proof The celebrated "first main theoremö f V.A.Vassiliev [41], see also [42], is a * *wonderful source of (established) h-principles. One of these Vassiliev h-principles, slig* *htly modified, turns out to be a rather close approximation to the one we are after in connect* *ion with the Mumford conjecture. We describe this approximation. Fix d 0 as before. For smooth X without boundary, let W(X) be the set of all * *pairs (ß, f) where ß :E ! X is a smooth submersion with oriented fibers of dimension * *d + 1, and f :E ! R is a smooth function subject to two conditions: the restriction of* * f to each fiber of ß is a Morse function, and (ß, f): E ! (X, R) is proper. Let hW(X) be the set of all pairs (ß, ^f) where ß :E ! X is a smooth submersion* * with fibers of dimension d + 1 as before, and f^is a section of the vertical 2-jet b* *undle p2i:J2i(E, R) -! E , subject to two conditions. Namely, for x 2 X and z 2 Ex = ß-1(x), the value f^(* *z) can be represented by a germ near z of Morse functions Ex ! R; and (ß, f): E ! X x * *R is proper, where f :E ! R is the underlying map of f^. An element (ß, f) 2 W(X), with ß :E ! X , determines a section j2if of the proj* *ection J2i(E, R) ! E by fiberwise 2-jet prolongation. The map W(X) -! hW(X) ; (ß, f) 7! (ß, j2if) (1.* *10) respects the concordance relation and so induces a map between the sets of conc* *ordance classes, W[X] ! hW[X]. This is obviously very similar to (1.9). The contravaria* *nt functors X 7! W[X] and X 7! hW[X] are representable, so that elements of W[X] and hW[X] * *are in bijective natural correspondence with homotopy classes of maps X -! |W| , X -! |hW| , 8 respectively, for certain spaces |W| and |hW|. The natural map W[X] ! hW[X] giv* *en by 2-jet prolongation corresponds a map between the representing spaces, |W| -! |hW| . (1.* *11) Vassiliev's first main theorem is the main ingredient in our proof of: Theorem 1.3.1 The map (1.11) is a homotopy equivalence. We now relate (1.11) to (1.9). The functors X 7! V[X] and X 7! hV[X] also ha* *ve representing spaces |V| and |hV|, respectively. The inclusions V(X) ! W(X) res* *pect the concordance relation and so induce a map |V| ! |W|. To have a similar incl* *usion hV(X) ! hW(X), we need to adjust the definition of hV(X) by insisting on pairs * *(ß, ^f), with ß :E ! X etc., where f^is a section of the fiberwise 2-jet bundle j2i:J2i(E, R) -! E , but the conditions are, as before, on the induced section of the fiberwise 1-je* *t bundle j1i. Then there is a commutative square |V|_____//_|hV| (1.* *12) | | | | fflffl| fflffl| |W| _____//|hW| where the vertical arrows are inclusion-induced and the horizontal ones are giv* *en by jet pro- longation. In order to make an efficient comparison between the two rows of dia* *gram (1.12), we introduce a öl calized" version of the second row. For a smooth X without boundary, let Wloc(X) consist of pairs (ß, f), with ß :E* * ! X etc., as in the definition of W(X), except for one change. We no longer requir* *e that (ß, f): E ! X x R be proper; instead we require that the restriction of (ß, f) * *to the fiberwise singularity set (ß, f) = {z 2 E | df = 0 on (T iE)z} be a proper map (ß, f) ! X x R. There is an h-version hWloc(X), consisting of * *pairs (ß, ^f) as in the definition of hW(X), except for a weakening of the properness* * condition. Jet prolongation defines a map between the representing spaces, |Wloc| ! |hWloc* *|. Theorem 1.3.2 The jet prolongation map |Wloc| ! |hWloc| is a homotopy equiva* *lence. This is much easier than 1.3.1. We briefly describe the ideas involved. For a* *n element (ß, f) 2 Wloc(X), the set = (ß, f) E consists of all z 2 E where f restric* *ted to the fiber of ß through z has a singularity. Since these singularities are nonde* *generate by assumption, is a smooth submanifold of E which is everywhere transverse to th* *e fibers of ß . Hence the projection ! X is an 'etale map, alias codimension zero sub* *mersion. 9 Because of the weakened properness condition, knowledge of the 'etale map ! X* * , the normal bundle of in E and the Morse index map ! {0, 1, 2, 3, . .,.d + 1} tu* *rns out to be sufficient to reconstruct the concordance class of (ß, f). This makes it * *easy to give a simple description of Wloc[X]. There is a similar description of hWloc[X], and * *theorem 1.3.2 is an easy consequence of these simplified descriptions. The sets hV(X) hW(X) hWloc(X) consist of smooth maps (ß, f): E ! X x R with extra tangential structure. It is always a relatively easy matter to class* *ify tangential structures, and in fact we are able to determine the homotopy type of all three* * spaces. The space |hV| was implicitly determined in the previous subsection: |hV| ' 1 hV . A very similar analysis gives |hW| ' 1 hW for another (twice looped down) Thom spectrum hW. Finally, Y |hWloc| ' 1 S1 (S1 ^ BSO (p, q)+ ) p,q p+q=d+1 where SO(p, q) GL (p + q) is the subgroup which stabilizes the standard quadr* *atic form (x1, . .,.xd+1) 7! -(x21+ . .+.x2p) + (x2p+1+ . .+.x2d+1) . Given the homotopy types, one observes the following Theorem 1.3.3 The maps |hV| ! |hW| ! |hWloc| define a homotopy fibration seq* *uence of infinite loop spaces. Theorems 1.3.1, 1.3.2, 1.3.3 are valid for any choice of d 0. This is not the* * case for the final result that goes into the proof of theorem 1.1.1, although substantial parts of* * it are valid for all d. For the moment we take d = 2. In this case we have a modified versio* *n Vc(X) of V(X). Namely, an element of Vc(X) is a proper submersion, or equivalently, a* * bundle (ß, f): E ! XxR whose fibers F are connected, closed, smooth, 2-dimensional; in* * addition we assume that each fiber comes equipped with an orientation preserving embeddi* *ng of S0 x D2. The functor X 7! Vc[X]`has a representing space |Vc|. By the discussio* *n leading up to 1.4, we have |Vc| ' gB g,2and therefore B|Vc| ' B 1+x Z . It is a consequence of theorems 1.3.1 and 1.3.2 that the spaces |W| and |Wloc| * *are group complete, so that |W| ' B|W| , |Wloc| ' B|Wloc| . 10 Theorem 1.3.4 With d = 2, the sequence B|Vc| -! |W| -! |Wloc| is a homotopy fibration sequence. Using this in conjunction with 1.3.1 and 1.3.2, we have another homotopy fibrat* *ion sequence B|Vc| ! |hW| -! |hWloc|, and therefore by theorem 1.3.3 the conclusion B 1+x Z ' B|Vc| ' |hV| ' 1 hV = 1 CP1-1. The proof of theorem 1.3.4 is technically the most demanding part of the paper.* * It rests on compatible stratifications of |W| and |Wloc|, where the strata are certain bund* *le theories. This part of the analysis is valid for all d 0. But the Harer stability theor* *em is also used in an essential way, if only as a black box. This leads to the condition d = 2. We end the paragraph with an explanation of where the stratifications of |W| an* *d |Wloc| come from, again for arbitrary but fixed d 0. Let (ß, f) be an element of W(X), with ß :E ! X and f :E ! R. We can then assoc* *iate to each x 2 X a finite set Tx. This is the set of critical points with critical* * value 0 of the Morse function f|Ex, where Ex = ß-1(x). It comes with a map Tx ! {0, 1, 2, . .,.d + 1} , the Morse index map. Therefore (ß, f) determines a partition of X into locall* *y closed subsets X, indexed by the isomorphism classes of finite sets over {0, 1* *, . .,.d + 1}. Namely, X X consists of all x 2 X such that Tx ~=T . If the partition has only one nonempty part corresponding to a single isomorphi* *sm class , then we say that (ß, f) is pure of class . At the other extreme, we h* *ave the case where (ß, f) is "generic". Then the partition of X determined by (ß, f) is a st* *ratification. Each stratum X is a smooth submanifold of X of codimension |T |. The image o* *f (ß, f) in W(X) is pure of class . Let W(X) W(X) consist of the elements which are pure of class . Divid* *ing by the concordance relation, we have a contravariant functor X 7! W[X]. The* * above observations suggest that the representing space |W| of X 7! W[X] has a stratif* *ied model whose strata are indexed by isomorphism classes of finite sets over {0, 1,* * . .,.d + 1}, and such that the stratum corresponding to is a representing space for X 7* *! W[X]. We confirm this in section 5. There is a compatibly stratified model of |Wloc|. The usefulness of the stratifications of |W| and |Wloc| comes from the fact tha* *t the strata |W| and |Wloc,| represent genuine bundle theories. To make this more prec* *ise in the case of |W|, let (ß, f) be an element of W(X). By definition, the projection* * from 0(ß, f) = (ß, f) \ f-1 (0) to X is then a |T |-sheeted covering. Consequently 0(ß, f) is a codimension 0 * *submanifold of (ß, f), hence a union of connected components of (ß, f). It turns out tha* *t the 11 remaining components of (ß, f) are "removable". That is, every class in W[X* *] has a representative (ß, f) with 0(ß, f) = (ß, f) . In this situation, ß :E ! X is automatically a bundle of (d + 1)-manifolds. Mo* *reover, for every nonzero c 2 R, the restriction of ß to f-1 (c) is a bundle of closed * *d-manifolds. When d = 2, this brings us back to surface bundles and leads (with the Harer st* *ability theorem) to a proof of theorem 1.3.4. * * * For the rest of this paper we consider only the surface case d = 2, with additi* *onal boundary data in the definitions of V , W etc., cf. section 2. The reader can easily ada* *pt the arguments of sections 2-5 to the general case d 2. 2 Some generalized bundle theories This section defines the six generalized bundle theories sketched out in the in* *troduction. They are considered to be sheaves on the category X of smooth manifolds withou* *t bound- ary (of arbitrary dimension) and with smooth maps as morphisms. 2.1 The basic sheaves Let E be a smooth manifold with boundary and ß :E ! X a smooth map to an object* * of X . The map ß is a submersion if its differentials dß : (T E)z ! T Xi(z), z 2 E r @E d(ß|@E): (T @E)z ! T Xi(z), z 2 @E are all surjective. In all cases considered below ß :E ! X is assumed to be a * *product bundle near @E . If the submersion ß is also proper, then ß is a smooth fiber b* *undle by Ehresmann's fibration theorem [2, thm. 8.12]. Pull-back or base change of bundles or submersions is not strictly associative.* * To get around this we shall assume that ß is a graphic map. The rule which to an X in X asso* *ciates the set of graphic submersions ß :E ! X is a contravariant functor on X . Definition 2.1.1 Let f :S ! T be a map of sets. We say that f is graphic if f * *has a factorization S ! U x T ! T where the first arrow is an inclusion (not just an * *injection) and the second arrow is the projection from U x T to T . 12 An arbitrary map f :S ! T has a graphic replacement f~:S~! T where ~S S x T is* * the graph of f . Let f :S ! T2 be a graphic map and let g :T1 ! T2 be any map. We make a pullback square g*S _____//S (2* *.1) | | | |f fflffl|gfflffl| T1 _____//_T2 by letting g*S consist of all ordered pairs (u, t) such that t 2 T1 and (u, g(t* *1)) 2 S . The left hand vertical arrow is given by (u, t) 7! t and it is again a graphic map.* * Moreover, if g is an identity, then g*S = S ; and if g is a composition, g = g2g1, then g*S * *= g1*g2*S . Thus, with the above definitions, base change is associative. Let E be a smooth manifold and pk: Jk(E, R) ! E the k-jet bundle, where k 0. * *Its fiber at z 2 E consists of equivalence classes of smooth map germs f :(E, z) ! * *R, with f equivalent to g if the k-th Taylor expansions of f and g agree at z (in local c* *oordinates near z ). The elements of Jk(E, R) are called k-jets of maps from E to R. The* * k-jet bundle pk: Jk(E, R) ! E is a vector bundle. A smooth function f :E ! R induces a smooth section jkf of pk, which we call th* *e k-jet prolongation of f , following e.g. Hirsch [19]. (Some writers choose to call it* * the k-jet of f , which can be confusing.) Not every smooth section of pk has this form. Sect* *ions of the form jkf are called integrable. Thus a smooth section of pk is integrable if an* *d only if it agrees with the k-jet prolongation of its underlying smooth map f :E ! R. We need a fiberwise version Jki(E, R) of Jk(E, R), fiberwise with respect to a * *submersion ß :Ej+r ! Xj with fibers Ex for x 2 X . In a neighborhood of any z 2 E we may c* *hoose local coordinates Rj x Rr so that ß becomes the projection onto Rj and z = (0, * *0). Two smooth map germs f, g :(E, z) ! R define the same element of Jki(E, R)z if thei* *r k-th Taylor expansions in the Rr coordinates agree at (0, 0). Thus Jki(E, R)z is a q* *uotient of Jk(E, R)z and Jki(E, R)z = Jk(Ei(z), R). There is a surjection of vector bundle* *s on E , Jk(E, R) -! Jki(E, R). Sections of the bundle projection pki:Jki(E, R) ! E will be denoted ^f, ^gand t* *heir under- lying smooth functions from E to R by f , g , etc. A smooth function f :E ! R induces a section jkif of pki, which we call the fib* *erwise k-jet prolongation of f . The sections of the form jkif are called integrable. We now take k = 2 and introduce the following (standard) Notation 2.1.2 (i) A section ^fof p2iis fiberwise nonsingular if ^f(z) 2 J2(E* *i(z), R) has a non-vanishing linear part, for each z 2 E . 13 (ii) A section f^of p2iis fiberwise Morse if each value f^(z) is either nonsing* *ular or, when singular, has a non-degenerate quadratic part. (iii) A smooth map f :E ! R is fiberwise nonsingular, resp. fiberwise Morse, if* * j2if is fiberwise nonsingular, resp. Morse. (iv) The singularity set (ß, ^f) E is the set of points z with ^f(z) singul* *ar. If ^f= j2if , then we write (ß, f) instead of (ß, ^f). Let i(E, R) J2i(E, R) be the submanifold consisting of the singular jets, i.* *e., those with vanishing linear part. Then for a section f^of p2i, we have (ß, ^f) = ^f-1( i(E, R)) , so that f^is fiberwise nonsingular if and only if it misses i(E, R). For integ* *rable f^, we can also say that f^is fiberwise Morse if and only if it is fiberwise transvers* *e to i(E, R). See [12, II.6.1-4]. This has the following consequence. Lemma 2.1.3 Suppose that f :E ! R is fiberwise Morse. Then the restriction o* *f ß to (ß, f) is a local diffeomorphism (ß, f) ! X . Proof The assumption implies that the fiberwise differential dif viewed as a * *section of the vertical cotangent bundle Ti*E ! E is transverse to the zero section. In p* *articular = (ß, f) is a submanifold of E , of the same dimension as E . But moreover,* * the fiberwise Morse condition implies that for each z 2 , the tangent space (T )* *z has trivial intersection in (T E)z with the vertical tangent space (T iE)z. This means tha* *t is transverse to each fiber of ß , and also that ß| is a local diffeomorphism. It is customary to call local diffeomorphisms for 'etale maps. We will follow t* *his tradition: Definition 2.1.4 A smooth map p: Y ! X between smooth manifolds of the same di- mension is called 'etale if its differential at every point y 2 Y is a linear * *isomorphism from (T Y )y to T Xi(z). Recall from Ehresmann's fibration lemma, [2, 8.12], that a proper submersion is* * a fiber bundle, and in particular that a proper 'etale map is a covering projection. We now define a number of sheaves on the category X , that is, contravariant fu* *nctors F from X to the category of sets, which satisfy the following condition: for eve* *ry X in X and open cover {Yi}i2I of X , the sequence Q ____//_Q ? ____//_F(X)____//i2IF(Yi) ____//_i,j2IxIF(Yi\ Yj) is exact. In other words, given si2 F(Yi) for i 2 I such that si|Yi\ Yj = sj|Yi* *\ Yj, there exists a unique s 2 F(X) with s|Yi= si. 14 Definition 2.1.5 For an object X in X , let hV(X) be the set of pairs (ß, ^f)* * where ß :E ! X is a graphic submersion with oriented 3-dimensional fibers and ^fis a * *section of p2i:J2i(E, R) ! E , subject to the following three conditions: (i) (ß, f): E ! X x R is proper. (ii) ^fis nonsingular. (iii) near their respective boundaries, E and (S1 x R x [0, 1] ) x X agree. Mor* *eover ß is the standard projection to X and f^is the jet prolongation of the project* *ion to R. Analogously we have an integrable version V(X) hV(X), consisting of pairs (ß,* * f) where f :E ! R is a smooth function such that (ß, j2if) 2 hV(X)). This means that for* * each x 2 X the restriction of f to the fiber Ex is nonsingular. Thus (ß, f): E ! X x* * R is a proper submersion, hence a smooth fiber bundle. Each fiber is an oriented surfa* *ce with two boundary circles, but it need not be connected. Definition 2.1.6 Let Vc(X) V(X) denote the subset of pairs (ß, f) where the * *fibers of (ß, f): E ! X x R are connected. Definition 2.1.7 For X in X let hW(X) be the set of pairs (ß, ^f), as in defi* *nition 2.1.5, which satisfy conditions (i) and (iii), but where condition (ii) is replaced by* * the weaker condition (iia) ^fis fiberwise Morse. Again we have an integrable version W(X) hW(X) consisting of pairs (ß, f) whe* *re f :E ! R is a smooth function such that (ß, j2if) 2 hW(X). The integrability co* *ndition means that, for each x 2 X , the restriction of f to the fiber Ex is a Morse fu* *nction. Definition 2.1.8 For X in X let hWloc(X) be the set of pairs (ß, ^f), as in * *defini- tion 2.1.5, which satisfy condition (iii), but where conditions (i) and (ii) ar* *e replaced by (ia) the map (ß, ^f) ! X x R ; z 7! (ß(z), f(z)) is proper, (iia) ^fis fiberwise Morse. The integrable version of hWloc(X) is denoted Wloc(X). Making X into a variabl* *e, we have contravariant functors hV , hW , hWlocon X and their integrable versions * *V , W , Wloc. This uses base change for graphic maps to X (here smooth submersions ß :E* * ! X ) as defined in 2.1.1. All six functors have the sheaf property. They fit toget* *her into the diagram of sheaves V ______//W ______//Wloc (2* *.2) j2ß|| j2ß|| j2ß|| fflffl| fflffl| fflffl| hV ____//_hW _____//hWloc. 15 2.2 Homotopy theory of sheaves Given a sheaf F on X we want to consider for each X in X the concordance clas* *ses of elements of F(X). This requires that we extend F to be defined on manifold* *s with boundary. It suffices to consider manifolds with collared boundary. For such X * *we have a canonical projection p: Y r @X ! @X for a sufficiently small open neighborhoo* *d Y of @X in X . Definition 2.2.1 Let X be smooth with collared boundary. Let F(X) be the set o* *f all pairs (r, s) 2 F(X r@X)xF(@X) such that r and p*(s) agree on Y r@X , for a suff* *iciently small open neighbourhood Y of @X in X . (Here p is the collar projection.) Definition 2.2.2 Let F be a sheaf on X , let X be an object of X and let s0, s* *1 2 F(X). A concordance from s0 to s1 is an element (h, s) of F(X x [0, 1]) such that s 7* *! (s0, s1) under the canonical bijection F(X x @[0, 1]) ! F(X) x F(X). We also say that h 2 F(Xx ]0, 1[ ) is a concordance from s0 to s1. If such a co* *ncordance exists, then s0 and s1 are said to be concordant and we write s0 ' s1 or h: s0 * *' s1. Concordance is an equivalence relation on F(X). The set of equivalence classes* * will be denoted by F[X]. It is necessary to have a relative version of F[X]. Suppose that A X is a clo* *sed subset, where X is in X . Let s 2 colimUF(U) where U ranges over the open neighborhoo* *ds of A in X . Note for example that any z 2 F(?) gives rise to such an element, * *namely s = {p*U(z)} where pU :U ! ? . In this case we often write z instead of s. Definition 2.2.3 Let F(X, A; s) F(X) consist of the elements t 2 F(X) whose * *germ near A is equal to s. Two such elements t0 and t1 are concordant relative to A * *if they are concordant by a concordance whose germ near A is the constant concordance from * *s to s. The equivalence classes are denoted F[X, A; s]. We now construct the representing space |F| of F and list its most important pr* *operties. Let be the category whose objects are the ordered sets n_:= {0, 1, 2, . .,.n}* * for n 0, with order preserving maps as morphisms. For n 0 let ne Rn+1 be the extend* *ed standard n-simplex, ne:= {(x0, x1, . .,.xn) 2 Rn+1 | xi= 1}. An order-preserving map m_ ! n_induces a map of affine spaces me! ne. This ma* *kes n___7! neinto a covariant functor from to X . Definition 2.2.4 The representing space |F| of a sheaf F on X is the geometri* *c realiz- ation of the simplicial set n_7! F( ne). 16 An element z 2 F(?) gives a point z 2 |F| and F[?] = ß0|F|. In appendix A we p* *rove that |F| represents the contravariant functor X 7! F[X]. Indeed we prove the fo* *llowing slightly more general Proposition 2.2.5 For X in X , let A X be closed and z 2 F(?). There is a n* *atural bijection # from the set of homotopy classes of maps (X, A) ! ( |F|, z) to the * *set F[X, A; z]. Taking X = Sn and A equal to the base point, we see that the homotopy group ßn(* *|F|, z) is identified with the set of concordance classes F[Sn, ?; z]. We introduce the* * notation ßn(F, z) := F[Sn, ?; z] . A map v :E ! F of sheaves induces a map |v|: |E| ! |F| of representing spaces. * *We call v a weak equivalence if |v| is a homotopy equivalence. Proposition 2.2.6 Let v :E ! F be a map of sheaves on X . Suppose that v indu* *ces a surjective map E[X, A; s] -! F[X, A; v(s)] for every X in X with a closed subset A X and a germ s 2 colimUE(U), where U ranges over the neighborhoods of A in X . Then v is a weak equivalence. Proof The hypothesis implies easily that the induced map ß0E ! ß0F is onto an* *d that, for any choice of base point z 2 E(?), the map of concordance sets ßn(E, z) ! ß* *n(F, v(z)) induced by v is bijective. * * __|_| Applying the representing space construction to the sheaves displayed in diagra* *m (2.2), but using Vc instead of V , we get a commutative diagram of representing spaces |Vc| _____//_|W| _____//_|Wloc| (2* *.3) j2ß|| j2ß|| j2ß|| fflffl| fflffl| |fflffl |hV| _____//|hW| _____//|hWloc|. a1 Note that |Vc| ' B g,2. g=0 17 2.3 Different models and monoid structures Let F be one of the sheaves from section 2.1. Concatenation along a boundary co* *mponent defines a composition law F[X] x F[X] -! F[X] so that each of the spaces in diagram (2.3) comes equipped with a multiplicatio* *n which is homotopy associative and with a homotopy unit. Our first task is to give an * *upgraded version of the sheaves that turns their values into monoids, and therefore make* *s the rep- resenting spaces into topological monoids (without unit). We describe this in d* *etail for W and leave the other cases to the reader. Let t: X !] 0, 1 [ be a smooth function. We define [0, t] x X := {(s, x) 2 R x X | 0 s t(x)}. Definition 2.3.1 For X in X let W0(X) be the set of quadruples (t, u, ß, f) w* *here t is a function as above, (u, ß): E ! [0, t] x X is a smooth graphic map whose X -co* *ordinate is a submersion ß :E ! X with 3-dimensional fibers, and f :E ! R is a smooth ma* *p, subject to the following conditions. (i) (ß, f): E ! X x R is proper. (ii) For each x 2 X the restriction fx: Ex ! R is Morse. (iii) Near their respective boundaries, the manifolds E and (S1 x R) x ( [0, t]* * x X) agree, and u, ß , f agree there with the obvious projections. There is a monoid structure on W0(X). Indeed, for (t, u, ß, f) and (t0, u0, ß0,* * f0) in W0(X) one defines (t, u, ß, f) O (t0, u0, ß0, f0) = (t + t0, u00, ß00, f00).* *(2.4) Here the source of ß00is the union (concatenation) of E and oe*E0, where oe* de* *notes the base change, as in (2.1), along the translation homeomorphism oe :[t, t + t0] x-X! [0, t0] x X ; (s, x)7! (s - t(x), x) . The maps ß00and f00are defined by ß00 = ß [ oe*ß0, f00 = f [ oe*f0. There is no unit for the product in (2.4), since we assumed t > 0 in definition* * 2.3.1. As a result the representing space |W0| becomes a topological monoid without a str* *ict unit. However, the classifying space construction B|W0| and hence the group completio* *n B|W0| make perfectly good sense. We also describe a way to attach an artificial unit * *to monoids without unit in appendix C. Lemma 2.3.2 The sheaves W0 and W are homotopy equivalent. 18 Proof There is a subsheaf W00 of W0 which we obtain by allowing only the cons* *tant function t with value 1 in definition 2.3.1. The inclusion W00! W0 is a weak eq* *uivalence, and so is the forgetful map from W00to W . * * __|_| There are similar enlarged models for the other sheaves of section 2.1, so diag* *ram (2.3) is equivalent to a diagram of topological monoids and monoid maps. In the rest of * *the paper, we will not make explicit use of these larger models with monoid structures: it* * is usually enough to know that they exist. We next introduce sheaves W0 and hW0 on X . They are weakly equivalent to W and hW , respectively, but are better related to Vassiliev's h-principle, see [42, * *Thm.0.A] and [41, III, 1.1], than W and hW . Definition 2.3.3 For X in X let hW0(X) be the set of all pairs (ß, ^f) as in * *defini- tion 2.1.7, replacing however condition (iia)by the weaker (iib) f^is fiberwise Morse in some neighborhood of f-1 (0). From the definition, there is an inclusion hW ! hW0. There are also an integrab* *le version W0 and an inclusion W ! W0. Lemma 2.3.4 The inclusions W ! W0 and hW ! hW0 are homotopy equivalences. Proof We will concentrate on the first of the two inclusions, W ! W0. Fix (ß* *, f) in W0(X), with ß :E ! X and f :E ! R. We will subject (ß, f) to a concordance endi* *ng in W(X). Choose an open neighborhood U of f-1 (0) in E such that, for each x 2 * *X , the critical points of fx = f|Ex on Ex \ U are all nondegenerate. Since E r U is cl* *osed in E and the map (ß, f): E ! X x R is proper, the image of E r U under that map is a* * closed subset of X x R which has empty intersection with X x 0. (Proper maps between l* *ocally compact spaces are closed maps.) We can therefore choose a smooth function ': X* * ! ]0, 1] such that U contains all z 2 E for which |f(z)| < '(ß(z)). And we can choose a * *smooth isotopy of embeddings 't: X x R ! X x R, where 0 t 1, such that (i) each 't is a map over X , (ii)'0 = idand '1(X x R) {(x, t) | -'(x) < t < '(x)}, (iii)'t= '0 for t close to 0 and 't= '1 for t close to 1. Then let E(t)be the inverse image of 't(X x R) under the map (ß, f): E ! X x R.* * Let f(t):E(t)! R be the second coordinate of 't-1following on (ß, f): E ! X x R. Le* *t ß(t) be the restriction of ß to E(t). Now t 7! (ß(t), f(t)) defines a concordance from (ß, f) 2 W0(X) to an element in W(X). (Strictly spe* *aking, some renaming of some of the elements of E(t)for 0 t 1 is required because * *of the 19 boundary conditions in the definitions.) If the restriction of (ß, f) to an ope* *n neighborhood Y1 of a closed A X belongs to W(Y1), then the concordance can be made relativ* *e to Y0, where Y0 is a smaller open neighborhood of A in X . * * __|_| For later use we list Lemma 2.3.5 If X in X is compact, then every class in W[X] or hW[X] has a r* *epres- entative (ß, f), resp. (ß, ^f), in which f :E ! R is a bundle projection, so th* *at E ~=f-1 (0) x R . Proof We concentrate on the first case. First pick a representative (ß, f) 2 * *W(X) such that f :E ! R has 0 as a regular value. Since (ß, f): E ! X x R is proper and X is compact, f itself is proper and therefore closed. It follows that there exi* *sts an " > 0 such that all s 2 [-", "] are regular values for f . Now choose an isotopy of * *embeddings 't: R ! R, where 0 t 1, such that '0 is the identity and '1(R) [-", "]. H* *ere t runs from 0 to 1 and the isotopy is stationary near t = 0 and t = 1. The shrinking a* *rgument of the previous lemma gives a concordance from (ß, f) to an element (ß(1), f(1)), * *where the source E(1)of ß(1)is f-1 ('1(R)) and f(1)equals f|E(1)followed by the inverse o* *f '1. Since f|E(1)is regular, f(1)is regular. Since f(1)is also proper, f(1)is a proper sub* *mersion, i.e., a bundle projection. * * __|_| 3 The spaces of diagram (2.3) This section determines the homotopy types of the spaces of (2.3), save the spa* *ce |W| which is deferred to section 4. 3.1 A cofiber sequence of Thom spectra Let Gr W (R3+n) be the manifold of triples (V, `, q) consisting of an oriented * *3-dimensional linear subspace V R3+n , a linear map `: V ! R and a quadratic form q :V !* * R, subject to the condition that if ` = 0, then q is nondegenerate. GrW (R3+n) cl* *assifies 3- dimensional oriented vector bundles whose fibers have the above extra structure* *, i.e., each fiber V comes equipped with a Morse type map `+q :V ! R and with a linear embed* *ding into R3+n . Let S3(R) be the vector space of quadratic forms on R3 (or equivalently, symmet* *ric 3 x 3 matrices) and S3(R) the subspace of the degenerate forms (not a linear subs* *pace). The complement Q(R3) = S3(R) r is the space of non-degenerate quadratic forms* * on R3, and A2(R3) = (R3)* x S3(R) r (0 x ) 20 is precisely the space of pairs (`, q) as above, where ` + q :R3 ! R is a Morse* * type map. The group GL (R3) acts on the right of A2(R3) by (`, q) . g = (`g, qg). Restri* *cting this action to SO (3) we have 2 3 ffi GrW (R3+n) ~= O(3 + n)=O(n) x A (R ) SO (3) . (3* *.1) We turn to a description of the homotopy type of A2(R3) in more familiar terms.* * Since quadratic forms can be diagonalized, a3 Q(R3) = Q(i, 3 - i) i=0 where Q(i, 3 - i) is the connected component containing the form qi given by qi(x1, x2, x3) = -(x21+ . .+.x2i) + (x2i+1+ . .+.x23). The stabilizer O(i, 3 - i) of qi for the (transitive) action of GL 3(R) on Q(i,* * 3 - i) has O(i) x O(3 - i) as a maximal compact subgroup and GL 3(R) has O(3) as a maximal compact subgroup. Hence the inclusion ffl (O(i) x O(3 - i)) O(3)- ! Q(i, 3 - i) ;coset ofg 7! qig is a homotopy equivalence, and therefore the subspace Q0(R3) = {q0, q1, q2, q3} . O(3) ~= ` 3i=0(O(i) x O(3 - i))fflO(3) (3* *.2) ~= ? q RP 2q RP 2q ? of Q(R3) is a deformation retract, Q(R3) ' Q0(R3). Lemma 3.1.1 There is a homotopy equivalence from the join S2 * Q0(R3) to A2(* *R3) which is equivariant for the actions of O(3). Proof The space A2(R3) is the union of ((R3)* - 0) x S3(R) and (R3)* x Q(R3) * *with intersection ((R3)* - 0) x Q(R3). Since S3(R) and (R3)* have canonical contrac* *tions and since the inclusion Q0(R3) ! Q(R3) is a homotopy equivalence, we get a cano* *nical homotopy equivalence from the double mapping cylinder (alias homotopy colimit) * *of the diagram S2 -! S2 x Q0(R3) -! Q0(R3) to A2(R3). The homotopy colimit is precisely the join S2 * Q0(R3) and the map r* *espects the O(3)-actions. * * __|_| The tautological 3-dimensional vector bundle UW,n on Gr W(R3+n) is canonically * *embedded in a trivial bundle Gr W (R3+n) x R3+n . Let U?W,n Gr W (R3+n) x R3+n 21 be the orthogonal complement, an n-dimensional vector bundle on Gr W (R3+n). The tautological bundle UW,n comes equipped with the extra structure consisting* * of a map from (the total space of) UW,n to R which, on each fiber of UW,n , is a Morse t* *ype map. (The fiber of UW,n over a point (V, q, `) 2 GrW (R3+n) is identified with the 3* *-dimensional vector space V and the map can then be described as ` + q .) For the submanifold W,n Gr W(R3+n) consisting of the triples (V, `, q) with * *` = 0 we have ffi W,n ~= O(3 + n)=O(n) x Q(R3) SO (3) . (3* *.3) The restriction of UW,n to W,n comes equipped with the extra structure of a fi* *berwise nondegenerate quadratic form. There is a canonical normal bundle for W,n in Gr* * W(R3+n) which is identified with U*W,n| W,n . Hence there is a homotopy cofiber sequence " * Gr V(R3+n) Ø____//GrW(R3+n) ____//_Th(UW,n| W,n) where Gr V(R3+n) = GrW (R3+n)r W,n and Th (. .).denotes the Thom space. This le* *ads to a homotopy cofiber sequence of Thom spaces Th (U?W,n|Gr V(R3+n)) -! Th (U?W,n) -! Th (U?W,n U*W,n| W,n) which, as n varies, becomes a homotopy cofiber sequence of spectra hV -! hW -! hWloc. Here we view Th (U?W,n) as the (2 + n)-th space of the spectrum hW, and similar* *ly for the other two spectra. We then have the corresponding infinite loop spaces 1 hV = colim 2+nTh (U?W,n|Gr V(R3+n)) , 1 hW = colim 2+nTh (U?W,n) , 1 hWloc = colim 2+nTh (U?W,n U*W,n| W,n). The homotopy cofiber sequence of spectra above yields a homotopy fiber sequence* * of infinite loop spaces 1 hV -! 1 hW -! 1 hWloc, (3* *.4) that is, 1 hV is homotopy equivalent to the homotopy fiber of the right-hand m* *ap. In particular there is a long exact sequence of homotopy groups associated with di* *agram (3.4) and a Serre-Leray spectral sequence of homology groups. Lemma 3.1.2 There is a homotopy equivalence of infinite loop spaces 1 hWloc' 1 S1+1 ( W,1 )+ where W,1 ' BSO (3) q BO(2) q BO(2) q BSO (3). 22 Proof Since UW,n| W,n comes equipped with a fiberwise nondegenerate quadratic* * form, U*W,n| W,n is canonically identified with UW,n| W,n . Consequently the restrict* *ion fi U?W,n U*W,nfi W,n fi is trivialized, so that Th (U?W,n U*W,nfi W,n) ' S3+n( W,n)+ . Hence 1 hWloc ' 1 S1+1 ( W,1 )+ S where W,1 = W,n . Using the description (3.3) of W,n and the homotopy equ* *ivalence Q(R3) ' Q0(R3), see (3.2), we get 0 3 ffi W,n ' O(3 + n)=O(n)) x Q (R ) SO (3). S The union nO(3+n)=O(n) is a contractible free SO (3)-space, so that W,1 is * *homotopy equivalent to the homotopy orbit space of the canonical right action of SO (3) * *on a3 ffl Q0(R3) ~= (O(i) x O(3 - i)) O(3) . __|_| i=0 The Grassmann manifold Gr2(R2+n) of oriented 2-planes P in R2+n can be identifi* *ed with a subspace of Gr V(R3+n) = GrW (R3+n) r W,n , by P 7! (R P, prR, 0). The inj* *ection is covered by a monomorphism of vector bundles fi L?n! U?W,nfiGrV(R3+n) where L?nis the standard n-plane bundle on Gr 2(R2+n). Lemma 3.1.3 The space Gr V(R3+n) is homotopy equivalent to SO (3+n)=SO (2)xS* *O (n), and the map Th (L?n) -! Th (U?W,n| GrV(R3+n)) just constructed is (2n + 1)-conn* *ected. Proof From (3.1) and lemma 3.1.1 we have an embedding and a homotopy equivale* *nce ffi O(3 + n)=O(n) x S2 SO (3) -! GrV (R3+n) ffi where O(3 + n)=O(n) x S2 SO (3) ~=O(3 + n)=(SO (2) x O(n)). Using this as an * *identific- ation, we may identify the above embedding Gr 2(R2+n) ! GrV (R3+n) with the inc* *lusion O(2 + n)=(SO (2) x O(n)) -! O(3 + n)=(SO (2) x O(n)) . This is (n + 1)-connected. Passing to the corresponding map between Thom spaces* * raises the connectivity by n. * * __|_| We collect the results of this section, 3.1, in Proposition 3.1.4 The homotopy fiber sequence (3.4) is homotopy equivalent to 1 CP1-1-! 1 hW -! 1 S1+1 ( W,1 )+ . __|_| 23 3.2 The spaces |hW| and |hV| In section 2.1 we described the jet bundle J2(E, R) and its fiberwise version a* *s certain spaces of smooth map germs (E, z) ! R, modulo equivalence. For our use in this * *section and the next it is better to view it as a construction on the tangent bundle. F* *or a vector space V , let J2(V ) denote the vector space of maps f^: V ! R , ^f(v)= c + `(v) + q(v) where c 2 R is a constant, ` 2 V * and q :V ! R is a quadratic map. This is a c* *ontra- variant continuous functor on vector spaces, so extends to a functor on vector * *bundles with J2(F )z = J2(Fz) when F is a vector bundle over E . When F = T E is the tangent bundle of a manifold E , then there is an isomorphi* *sm of vector bundles J2(T E) ~=J2(E, R). Indeed after choice of a spray [2] on E , the associated exponential map induce* *s a diffeo- morphism germ expz: (T Ez, 0) ! (E, z) and f^(z) 2 J2(T Ez) gives an element of J2(E, R)z. The resulting vector bundl* *e map J2(T E) ! J2(E, R) is the required isomorphism. (To see that it is smooth one m* *ay use that the exponential map takes a neighborhood of the zero section in T E diffeo* *morphically to a neighborhood of the diagonal in E x E .) Given a submersion ß :E ! X with vertical tangent bundle T iE , we similarly ha* *ve an isomorphism of vector bundles J2(T iE) ~=J2i(E, R) . (3* *.5) This time we need an exponential map T E ! E for which the restricted map T iE * *! E is a map over X , i.e., such that ((T iE)z, 0) ! (Ei(z), z) is a diffeomorphism ge* *rm for each z 2 E . Our object now is to construct a natural map ø :hW[X] -! [X, 1 hW] (3* *.6) compatible with the concatenation in the source and loop sum in the target. Her* *e [ , ] in the right-hand side denotes a set of homotopy classes of maps. We assume familiarity with the Thom-Pontryagin relationship between Thom spectr* *a and their infinite loop spaces on the one hand, and bordism theory on the other. S* *ee [39] and especially [31]. Applied to our situation this identifies [X, 1 hW] with * *a group of 24 bordism classes of certain triples (M, g, ^g). Here M is smooth without boundar* *y, dim(M) = dim (X) + 2, and g , ^gtogether constitute a vector bundle pullback square ^g j T M x R1+j _____//T X x UW,1 x R (3* *.7) | | | | fflffl|g fflffl| M ________//_X x GrW (R3+1 ). The X -coordinate of g is required to be a proper map M ! XS. (We write UW,1 f* *or the tautological 3-dimensional vector bundle on Gr W (R3+1 ) = rGrW (R3+r).) The * *sum of bordism classes is given by disjoint union of representatives. For our purposes a slightly different description is preferable. For this we f* *ix a triple (S1, g0, ^g0) in which g0: S1 ! GrW (R3) is the constant map to the base point * *? = (R3, `, 0) with `(t1, t2, t3) = t1, and ^g0is the composite map standard framingxid switch (T S1 x R) x R__________________//_R2 x R_________________//_R x R2. We can then describe [X, 1 hW] as the bordism group of triples (M, g, ^g) as a* *bove but with @M = S1x@[0, 1]xX . (The restrictions of g and ^gto @M and T M|@M , respec* *tively, are also prescribed: they must agree with the pullbacks of g0 and ^g0under the * *projection from S1 x @[0, 1] x X to S1.) With this description, the group structure is g* *iven by concatenation, much as in section 2.3. The isomorphism from the standard descri* *ption to the modified one is given by taking disjoint union with S1 x [0, 1] x X . Let now (ß, ^f) 2 hW(X), where ß :E ! X is a submersion with 3-dimensional fibe* *rs and f^ is a section of J2(T iE) ! E with underlying map f :E ! R. See definition 2.* *1.7. After a small deformation which does not affect the concordance class of (ß, ^f), we * *may assume that f is transverse to 0 2 R (not necessarily fiberwise) and get a manifold M * *= f-1 (0) with dim(M) = dim(X) + 2. The boundary @M is identified with S1 x @[0, 1] x X a* *nd the restriction of ß to M is a proper map M ! X , by the definition of hW(X). The s* *ection f^ yields for each z 2 E a map ^f(z)= f(z) + `z + qz: (T iE)z ! R with the property that the quadratic term qz is nondegenerate when the linear t* *erm `z is zero. For z 2 M the constant f(z) is zero, so the restriction T iE|M is a 3-dim* *ensional oriented vector bundle on M with the extra structure considered in subsection 3* *.1. Thus T iE|M is classified by a map from M to the space Gr W(R3+1 ): there is a bundl* *e diagram T iE|M ________//UW,1 | | | | fflffl|~ fflffl| M _______//GrW(R3+1 ). Let g :M -! X x GrW (R3+1 ) be the map z 7! (ß(z), ~(z)). We now have a canonic* *al vector bundle map ^g:T M x R ~= T E|M ~= ß*T X|M T iE|M -! T X x UW,1 25 and we get a triple (M, g, ^g) which represents an element of [X, 1 hW] in the* * (modified) bordism-theoretic description. It is easily verified that the bordism class of* * (M, g, ^g) de- pends only on the concordance class of the pair (ß, ^f). Thus we have defined t* *he map ø of 3.6. Theorem 3.2.1 The natural map ø :hW[X] ! [X, 1 hW] is a bijection when X is* * a closed manifold. Proof We define a map oe in the other direction by running the construction ø* * backwards. Again we view [X, 1 hW] as a bordism group. Let (M, g, ^g) be a representativ* *e, with g :M ! X x GrW (R3+1 ) and ^g:T M x R1+j -! T X x UW,1 x Rj. We also assume @M = S1 x @[0, 1] x X . By obstruction theory, see lemma 3.2.2 b* *elow, we can suppose that j = 0. Writing E = M x R we obtain a map ßE :E ! X by composing the projection E ! M with the first component of g . Similarly the map ^g, now * *with j = 0 and T M x R = T E , has a first component which is a map of vector bundles ^ßE:T E -! T X, covering ßE and epimorphic in the fibers. Since E is an open manifold, Phillips* *' submersion theorem [30], [15], [16] applies to show that (ßE , ^ßE) is homotopic through f* *iberwise sur- jective bundle maps to a pair (ß, dß) where ß :E ! X is a submersion and dß :T * *E ! T X is its differential. This homotopy lifts to a homotopy of (fiberwise isomorphic) vector bundle maps,* * starting with ^g:T E ! T X x UW,1 and ending with a map T E ! T X x UW,1 which refines the differential dß :T E ! T X . Its restriction to T iE T E is a vector bun* *dle map T iE ! UW,1 which equips each fiber (T iE)z of T iE with a Morse type map `z + qz: (T iE)z ! R. Let f :E ! R be the projection onto the R factor, and let ^f(z) = f(z) + `z + qz 2 J2(T iE) ~=J2i(E, R). The map f is proper, since X and hence M are compact. Consequently the pair (ß* *, ^f) represents an element in hW[X]. Its concordance class depends only on the bordi* *sm class of (M, g, ^g). This describes a map oe :[X, 1 hW] -! hW[X]. It is obvious from the constructions that ø O oe = id. In order to evaluate the* * composition oe O ø , it suffices by lemma 2.3.5 to evaluate it on an element (ß, ^f) where * *f :E ! R is regular, so that E ~=M x R with M = f-1 (0). For (y, r) 2 M x R, the map f^(y, r): (T i(M x R))(y,r)-! R 26 is a second degree polynomial of Morse type. The homotopy ^f(t)(y, r) = ^f(y, tr) + (1 - t)r shows that (ß, ^f) is concordant to (ß, ^f(0)), which represents the image of (* *ß, ^f) under oe O ø . Therefore oe O ø = id. * * __|_| Lemma 3.2.2 Let T and U be k-dimensional vector bundles over a manifold M .* * Let [T, U]isobe the set of homotopy classes of isomorphisms fl :T ! U . The stabil* *ization map [T, U]iso-! [T x R, U x R]isois bijective for k > dim (M) + 1 and surjectiv* *e for k = dim(M) + 1. Proof Let iso(T, U) ! M be the fiber bundle over M whose fiber at x 2 M is* * the space of linear isomorphisms from Tx to Ux. Then [T, U]isois the set of homotop* *y classes of sections of iso(T, U) ! M . The fibers of iso(T, U) ! M are homotopy equiva* *lent to O(k), and ßj(O(k + 1), O(k)) = 0 for j < k. Induction over the skeletons in a (* *smooth) triangulation of M completes the proof. * * __|_| Finally we give a short description of a map |hW| ! 1 hW which induces 3.6. We* * allow ourselves some flexibility with the models for |hW| and 1 hW. Fix an integer r > 0 and X in X . To the data (ß, ^f) in definition 2.1.7, with* * ß :E ! X and f :E ! R, we add the following: a smooth embedding w :E -! X x R x [0, 1] x R1+r which covers (ß, f): E ! X x R, and a vertical tubular neighborhood N for the * *sub- manifold w(E) of X x R x [0, 1] x R1+r, so that the projection N ! w(E) is a ma* *p over X x R. Near @E both w and N are assumed to be standard. In particular, near @E * *the embedding w must agree with the standard embedding of (S1 x R x [0, 1] ) x X~= X x R x [0, 1] x S1 in X x R x R1+rx [0, 1]. Making X into a variable now, we can interpret the for* *getful map taking (ß, ^f, w, N) to (ß, ^f) as a map of sheaves hW(r)-! hW on X . This map is highly connected if r is large, by Whitney's embedding theor* *em, so that the resulting map from colimrhW(r)to hW is a weak equivalence of sheaves. Let Z(r)be the sheaf taking an X in X to the set of pointed maps S1 ^ (X x R)+ -! 1+rTh (U?W,r) where we use an exotic base point in the target, to be specified below. Then th* *e representing space of Z(r)is a good approximation to 1 hW, that is, colimr|Z(r)| ' 1 hW. T* *he Thom-Pontryagin collapse construction gives us a map of sheaves ø(r):hW(r)-! Z(r). (3* *.8) 27 In detail: let (ß, ^f, w, N) be an element of hW(r)(X). We assume that f^ is * *a section of J2(T iE) ! E , see (3.5). Now the differential dw promotes each fiber (T iE)* *z of the vector bundle T iE to a triple (Vz, `z, qz) 2 GrW (R3+r). Here Vz = dw((T iE)z)* *, viewed as a subspace of the vertical tangent space at w(z) of the projection X x R x [0, 1] x R1+r-! X , which we in turn may identify with R3+n , and `z + qz is the non-constant part * *of f^(z). In particular z 7! (Vz, `z, qz) defines a map ~: E ! GrW (R3+r). This extends cano* *nically to a pointed map Th (N) -! Th (U?W,r) because N is identified with ~*U?W,r. But Th (N) is a quotient of X x R x [0, 1* *] x S1+r where we regard S1+r as the one-point compactification of R1+r. Thus we have co* *nstructed a map X x R x [0, 1] x S1+r -! Th (U?W,r) or equivalently, X x R x [0, 1] -! 1+rTh (U?W,r). This is constant on X x R x * *@[0, 1] by inspection, the constant value being the exotic base point. Therefore our map c* *an also be written in the form S1 ^ (X x R)+ -! 1+rTh (U?W,r) and so is an element of Z(r)(X). This element is the image of (ß, ^f, w, N) un* *der ø(r) in (3.8). Taking colimits over r, we therefore have a diagram |hW| oo'__colimr|hW(r)|_____//colimr|Z(r)|'_//_ 1 hW which we informally describe as a map ø :|hW| ! 1 hW. Theorem 3.2.3 The map ø :|hW| ! 1 hW is a homotopy equivalence. Proof This follows easily from theorem 3.2.1 and the fact that ø can be taken* * to be a map of topological monoids (cf. section 2.3). First, theorem 3.2.1 with X = ? i* *mplies that the map ø induces a bijection ß0|hW| -! ß0( 1 hW) and consequently that ß0|hW| is a group, like ß0( 1 hW). Next, we use theorem * *3.2.1 with X = Sn , noting that the grouplike monoid structures imply ffi ffi ßn|hW| ~= [Sn, |hW| ] [?, |hW| ] , ßn( 1 hW) ~= [Sn, 1 hW] [?, 1 hW] for arbitrary choices of base points. Thus the map ø induces an isomorphism of * *homotopy groups, and Whitehead's theorem implies that it is a homotopy equivalence. * * __|_| The arguments above work in a completely similar fashion to identify |hV|. In f* *act the map ø in theorem 3.2.3 restricts to a map from |hV| to 1 hV and the analogue of th* *eorem 3.2.1 holds. Keeping the letter ø for this restriction, we therefore have Theorem 3.2.4 The map ø :|hV| ! 1 hV is a homotopy equivalence. * * __|_| 28 3.3 The space |hWloc| We start with a description of [X, 1 hWloc] as a bordism group. This is very s* *imilar to the description of [X, 1 hW] used in the construction of the map (3.6). Lemma 3.3.1 For X in X , the group [X, 1 hWloc] can be identified with the * *group of bordism classes of triples (M, g, ^g) consisting of a smooth M without boundary* *, dim(M) = dim (X) + 2, and a vector bundle pullback square ^g j T M x R1+j _____//T X x UW,1 x R | | | | fflffl|g fflffl| M _________//X x GrW (R3+1 ) where the restriction of the X -coordinate of g to g-1(X x W,1 ) is proper. Proof We first identify UW,1 | W,1 with its dual using the canonical quadrat* *ic form q , and then with the normal bundle N of W,1 in Gr W (R3+1 ). Let (M, g, ^g) be a* * triple as above, but with g transverse to X x W,1 . Then Z = g-1(X x W,1 ) is a smoo* *th (n - 1)-dimensional submanifold of M , with normal bundle NZ . Restriction of g* * and ^g yields a vector bundle pullback square (T Z NZ ) x R1+j_____//_T X x N x Rj | | | | | | fflffl| |fflffl Z ______________//X x W,1 . But since NZ is also identified with the pullback of N , this amounts to a vec* *tor bundle pullback square ^gZ k T Z x R1+k_____//_T X x W,1 x R (3* *.9) | | | | | fflffl|gZ fflffl| Z _____________//X x W,1 for some k 0. Here the X -coordinate Z ! X of gZ is still a proper map. Conversely, given data Z , gZ and ^gZas in (3.9), let M be the (total space of * *the) pullback of N to Z . There is a canonical map from M to N GrW (R3+1 ), and another fro* *m M to X , hence a map g :M ! X x GrW (R3+1 ). Moreover ^gZdetermines the ^gin a tr* *iple (M, g, ^g) as above. In this way, the bordism group in 3.3.1 is isomorphic to * *the bordism group of triples (Z, gZ , ^gZ) as in (3.9). But this is the standard bordism gr* *oup description of [X, 1 hWloc]; see lemma 3.1.2. * * __|_| 29 We now turn to the construction of a localized version of (3.6), namely, a natu* *ral map ø loc:hWloc[X] -! [X, 1 hWloc]. (3.* *10) First we modify the bordism group description in 3.3.1 by requiring @M = S1 x @* *[0, 1] x X instead of @M = ;. The group structure is then given by concatenation. Let now (ß, ^f) 2 hWloc(X), where ß :E ! X is a submersion with 3-dimensional f* *ibers and f^is a section of J2(T iE) ! E with underlying map f :E ! R. See definition* * 2.1.8 and (3.5). We may assume that f is transverse to 0 and get a manifold M = f-1* * (0). Proceeding exactly as in the construction of the map (3.6), we can promote this* * to a triple (M, g, ^g) where (g, ^g) is a vector bundle pullback square ^g j T M x R1+j _____//T X x UW,1 x R | | | | fflffl|g fflffl| M ________//_X x GrW (R3+1 ) . This time, we cannot expect that the X -component of g , in other words ß|M , i* *s proper. But its restriction to g-1(X x W,1 ) = (ß, ^f) \ M is proper, thanks to condition (ia) in definition 2.1.8. Therefore (M, g, ^g) * *represents an element in [X, 1 hWloc]. This is the image of (ß, ^f) under ø loc. * * __|_| Theorem 3.3.2 The natural map ø loc:hWloc[X] ! [X, 1 hWloc] is a bijection. Proof There is a map oe locin the other direction. The construction of oe loc* *is analogous to that of oe in the proof of theorem 3.2.1. It is clear that ø locO oe locis t* *he identity. The verification of oe locO ø loc= iduses lemma 3.3.3 below. * * __|_| Lemma 3.3.3 Let (ß, ^f) 2 hWloc(X), with ß :E ! X . Let U be an open neighbo* *rhood of @E [ (ß, ^f) in E . Then (ß|U, ^f|U) 2 hWloc(X) is concordant to (ß, ^f). Proof The concordance that we need is an element (ß], ^f)]in hWloc(Xx ]0, 1[ * *). Let E] be the union of Ex ]0, 1=2[ and Ux ]0, 1[ . Let ß](z, t) = (ß(z), t) and f^](z,* * t) = (f^(z), t) for (z, t) 2 E]. Some renaming of the elements of E] is required to ensure tha* *t ß] is graphic. * * __|_| Next we give a short description of a map |hWloc| ! 1 hWlocwhich induces (3.10* *). This is analogous to the construction of the map named ø in theorem 3.2.3. Fix an integer r > 0 and X in X . To the data (ß, ^f) in definition 2.1.8, with* * ß :E ! X and f :E ! R, we add the following: a smooth embedding w :E -! X x R x [0, 1] x R1+r 30 which covers (ß, f): E ! X x R, a vertical tubular neighborhood N for the subma* *nifold w(E) of X x R x [0, 1] x R1+r, and a smooth function _ :E ! [0, 1] such that _(* *z) = 1 for all z 2 (ß, ^f). We require that the restriction of (ß, f): E ! X x R to t* *he support of _ be proper. Near @E , the function _ is assumed to vanish and both w and N* * are assumed to be standard. Making X into a variable now, we can interpret the forgetful map taking (ß, ^f,* * w, N, _) to (ß, ^f) as a map of sheaves hW(r)loc-! hWloc on X . This map is highly connected if r is large. Let Z(r)locbe the sheaf taki* *ng an X in X to the set of pointed maps ? 3+r ? S1 ^ (X x R)+ -! 1+rcone Th (UW,r|Gr V(R )) ,! Th (UW,r) . Here the cone is a reduced mapping cone, regarded as a quotient of a subspace of Th (U?W,r) x [0, 1] , with Th (U?W,r) x 1 corresponding to the base of the cone. The Thom-Pontryagin * *collapse construction gives us a map of sheaves ø(r)loc:hW(r)loc-! Z(r)loc. (3.* *11) In detail: let (ß, ^f, w, N, _) be an element of hW(r)loc(X). We assume that f* *^is a section of J2(T iE) ! E , see (3.5). The differential dw promotes each fiber (T iE)z of* * the vector bundle T iE to a triple (Vz, `z, qz) 2 GrW (R3+r), as in the proof of theorem (* *3.2.3). In particular the formula z 7! ((Vz, `z, qz), _(z)) defines a map ~: E ! GrW (R3+r* *) x [0, 1]. This fits into a vector bundle pullback square N ____^~__//U?W,rx [0, 1] | | | | | fflffl|~ fflffl| E _____//GrW(R3+r) x [0, 1] because N is identified with ~*U?W,r. Now we obtain a map from X x R x [0, 1] x* * S1+r to the mapping cone ? 3+r ? cone Th (UW,r|Gr V(R )) ,! Th (UW,r) , viewed as a subquotient of Th (U?W,r)x[0, 1], by z 7! ^~(z) for z 2 N and z 7! * *? for z =2N . It can also be written in the form ? 3+r ? S1 ^ (X x R)+ -! 1+rcone Th (UW,r|Gr V(R )) ,! Th (UW,r) so that it is an element of Z(r)loc(X). This defines the map ø(r)loc. Taking co* *limits over r, we therefore have a diagram |hWloc|oo'__colimr|hW(r)loc|__//colimr|Z(r)loc|'//_ 1 hWloc 31 which we informally describe as a map ø loc:|hWloc| ! 1 hWloc. The following * *is a straightforward consequence of theorem 3.3.2 (cf. the proof of theorem 3.2.3): Theorem 3.3.4 The map øloc:|hWloc| ! 1 hWlocis a homotopy equivalence. * * __|_| The combination of theorems 3.3.4, 3.2.3, 3.2.4 and proposition 3.1.4 amounts t* *o a proof of theorem 1.3.3 from the introduction. Remark 3.3.5 It must be understood that the expression öh motopy fiber seque* *nceü sed in theorem 1.3.3 is short for a commutative square of pointed spaces and maps X0 ____//_X1 | | | | fflffl| fflffl| C _____//X2 which is homotopy cartesian and has a contractible lower left-hand term C . Thi* *s leaves us with the task of saying exactly how the lower row of diagram (2.3) should be co* *mpleted to a commutative square in which the added term is contractible. Define a sheaf hVlocon X by copying definition 2.1.5, the definition of hV , b* *ut leaving out condition (i). Then |hVloc| is contractible by an application of propositio* *n 2.2.5. There is a commutative square of inclusion maps of pointed CW-spaces |hV| _______//|hW| (3.* *12) | | | | fflffl| |fflffl |hVloc|_____//|hWloc|. The precise meaning of theorem 1.3.3, apart from the statement |hV| ' 1 CP1-1,* * is that (3.12) is homotopy cartesian. That is also what we have proved. 3.4 The space |Wloc| The goal is to show that the inclusion of Wlocin hWlocis a weak equivalence. We* * begin with the observation that the analogue of lemma 3.3.3 holds for Wloc: Lemma 3.4.1 Let (ß, f) 2 Wloc(X), with ß :E ! X . Let U be an open neighborh* *ood of @E [ (ß, f) in E . Then (ß|U, f|U) 2 Wloc(X) is concordant to (ß, f). * * __|_| Corollary 3.4.2 For X in X , there are natural bijections between Wloc[X] and 32 (i) the set of bordism classes of triples ( , p, g), where is a smooth manif* *old without boundary, p: ! X x R is a proper smooth map whose X -coordinate ! X is an 'etale map (= codimension zero immersion), and g is a map from to W* *,1 ; (ii) the set of bordism classes of triples ( 0, v, c) where 0 is a smooth mani* *fold without boundary, v : 0 ! X is a proper smooth codimension 1 immersion with orien* *ted normal bundle and c is a map from 0 to W,1 . Proof An element (ß, f) of Wloc(X) determines by lemma 2.1.3 a triple ( , p, * *g) as in (i), where is (ß, f) and p(z) = (ß(z), f(z)) for z 2 E . The map g : ! * * W,1 classifies the vector bundle T iE| with the nondegenerate quadratic form deter* *mined by (one-half) the fiberwise Hessian of f . Conversely, given a triple ( , p, g) w* *e can make an element (ß, f) in Wloc(X). Namely, let U be the 3-dimensional vector bundl* *e on classified by g , with the canonical quadratic form q :U ! R. Let E be the dis* *joint union of U and (S1 x R x [0, 1] ) x X . Let (ß, f): E ! X x R agree with q + ~pon U* * , where ~p denotes the composition of the vector bundle projection U ! with p: ! X* * x R. The resulting maps from Wloc[X] to the bordism set in (i), and from the bordism* * set in (i) to Wloc[X], are inverses of one another: One of the compositions is obviously a* *n identity, the other is an identity by lemma 3.4.1. Next we relate the bordism set in (i) to that in (ii). A triple ( , p, g) as i* *n (i) gives rise to a triple ( 0, v, c) as in (ii) provided p is transverse to X x 0. In t* *hat case we set 0 = p-1(X x 0) and define v and c as the restrictions of p and g , respect* *ively. Conversely, a triple ( 0, v, c) as in (ii) does of course determine a triple ( * *, p, g) as in (i) with = 0 x R. The resulting maps from the bordism set in (i) to that in (ii)* *, and vice versa, are inverses of one another: One of the compositions is obviously an ide* *ntity, the other is an identity by a shrinking lemma analogous to (but easier than) lemma * *2.3.5. __|_| It is well known that the bordism set (ii) in corollary 3.4.2 is in natural bij* *ection with [X, 1 S1+1 ( W,1 )+ ]~= [X, 1 hWloc]. Namely, Thom-Pontryagin theory allows us to represent elements of [X, 1 S1+1 (* * W,1 )+ ] by quadruples ( 0, v, ^v, c) where 0 is smooth without boundary, v and ^vconst* *itute a vector bundle pullback square T 0 x R1+j__^v_//T X x Rj | | | | fflffl|v fflffl| 0 ____________//X (for some j 0) with proper v , and c is any map from 0 to W,1 . By lemma 3.* *2.2 we can take j = 0 and by immersion theory we can assume ^v= dv , that is, v is an * *immersion and ^vis its (total) differential. 33 Consequently Wloc[X] is in natural bijection with [X, 1 hWloc]. It is easy to * *verify that this natural bijection is induced by the composition " filoc |Wloc|Ø____//|hWloc|____// 1 hWloc where ø locis the map of (3.11), 3.10 and theorem 3.3.4. We conclude that the c* *omposition is a homotopy equivalence (cf. the proof of theorem 3.2.3). Since ølocitself is* * a homotopy equivalence, it follows that the inclusion |Wloc| ,! |hWloc| is a homotopy equi* *valence. This is theorem 1.3.2 from the introduction. 4 Application of Vassiliev's h-principle This section contains the proof of theorem 1.3.1. It is based upon a special ca* *se of Vassiliev's first main theorem, [41, ch.III] and [42]. Let A J2(Rr, R) denote the space of 2-jets represented by f :(Rr, z) ! R with* * f(z) = 0, df(z) = 0 and det(d2f(z)) = 0, where d2f(z) denotes the Hessian. This set has c* *odimension r + 2 and is invariant under diffeomorphisms Rr ! Rr. Let Nr be a smooth compact manifold with boundary and let _ :N ! R be a fixed s* *mooth function with j2_(z) =2A for z in a neighborhood of the boundary. (Use local co* *ordinates near z ; the condition means that near @N , all singularities of _ with value 0* * are of Morse type, i.e., nondegenerate.) Define spaces (N, A, _) = {f 2 C1 (N, R) | f = _ near @N, j2f(z) =2A forz 2 N}, h (N, A, _) = {f^2 J2(N, R) | ^f= j2_ near @N, ^f(z) =2A forz 2 N}, where J2(N, R) denotes the space of smooth sections of the jet bundle J2(N, R* *) ! N . Both are equipped with the standard C1 topology. The special case of Vassiliev* *'s theorem that we need is the statement that the map j2: (N, A, _) -! h (N, A, _) (4* *.1) induces an isomorphism on integral homology. We use this when dim (N) = 3. 4.1 Sheaves with category structure Let F :X ! C atbe a sheaf with values in small categories. Taking nerves define* *s a sheaf with values in the category of simplicial sets, NoF :X ! Setso with N0F the sheaf of objects, N0F = ob(F), and N1F the sheaf of morphisms. We * *have the associated bisimplicial set NoF( oe) and recall [32] that the realization o* *f its diagonal is homeomorphic to either of its double realizations, fi fifi fi | k_7! NkF( ke) |~= fi`_7! | k_7! NkF( `e)~|fi=fik_7! | `_7! NkF( `e) |fi* *.(4.2) 34 Since | `_7! NkF( `e) | = Nk|F| by A.2.1, the right hand side of (4.2) is the * *classifying space B|F| of the topological category |F|. We next give another construction of B|F| related to Steenrod's view of princip* *al bundles as 1-cocycles. We shall consider locally finite open covers Y? = (Yj)j2J of spa* *ces X in X indexed by a fixed uncountable set J . For each finite nonempty subset S J we* * write " YS = Yj. j2S Associated to the cover Y? there is a topological category, denoted XY? in [37,* * x4], with a a ob(XY?) = YS , mor(XY?) = YS . S R,S R S A continuous functor from XY? to a topological group G, viewed as a topological* * category with one object, is equivalent to a collection of maps 'RS :YS -! G , one for each pair R S of finite subsets of J , subject to the cocycle conditi* *ons listed in definition 4.1.1 below. More general versions can be found in [29]. Definition 4.1.1 For X in X an element of fiF(X) is a pair (Y?, '?) where Y? * *is a locally finite open covering of X , indexed by J , and '? associates to each pa* *ir of finite, nonempty subsets R S of J a morphism 'RS 2 N1F(YS) subject to the following c* *ocycle conditions: (i) every 'RR is an identity morphism; (ii) for R S T , we have 'RT = ('RS |YT) O 'ST . Condition (ii) includes the condition that the right-hand composition is define* *d; in partic- ular, taking S = T one finds that the source of 'RS is the object 'SS , and tak* *ing R = S one finds that the target of 'ST is 'SS|YT . Remark 4.1.2 By an open cover of X indexedSby J we mean a map j 7! Yj from J* * to the set of open subsets of X such that jYj = X . This map is not required to * *be injective, and cannot always be injective, as the case X = ; shows. In definition 4.1.1, w* *e use open covers indexed by a fixed set J to ensure that fiF has the sheaf property. In a* *ppendix B, definition B.1.1, we give a variant of definition 4.1.1 which does not mention * *an indexing set, but uses surjective 'etale maps to X rather than open covers of X . The sets fiF(X) define a sheaf fiF :X ! Sets and hence a space |fiF|. The follo* *wing key theorem is one of our main tools used in the proof of both theorem 1.3.1 and th* *eorem 1.3.4. It may be viewed as a generalization of the result that isomorphism classes of * *Steenrod's principal coordinate bundles (over X ) are in bijective correspondence with hom* *otopy classes of maps from X to BG. See also [29]. Its proof is deferred to appendix B. 35 Theorem 4.1.3 The spaces |fiF| and B|F| are homotopy equivalent. Definition 4.1.4 Let E, F :X ! Cat be sheaves and g :E ! F a map between them. We say that g is a transport projection if the following square is a pullback s* *quare of sheaves on X : d0 N1E ____//_N0E |g| g|| fflffl|d0 fflffl| N1F ____//_N0F where d0 is the source operator. Proposition 4.1.5 Let g :E ! F and g0: E0 ! F be transport projections as in * *defin- ition 4.1.4. Let u: E ! E0 be a map of sheaves over F which respects the cat* *egory structures. Suppose that the maps N0E ! N0F and N0E0 ! N0F obtained from g and * *g0 have the concordance lifting property, cf. definition A.2.4. Suppose also that,* * for each object a of F(?), the restriction Ea ! E0aof u to the fibers over a is a weak equivale* *nce (resp. induces an integral homology equivalence of the representing spaces). Then fiu:* * fiE ! fiE0 is a weak equivalence (resp. induces an integral homology equivalence of the re* *presenting spaces). Proof According to theorem 4.1.3 it suffices to prove that u induces a homoto* *py (homo- logy) equivalence from B|E| to B|E0|. By (4.2) it is then also enough to show t* *hat Nk(u): NkE -! NkE0 becomes a homotopy equivalence (homology equivalence) after passage to represen* *ting spaces, for each k 0. Since g and g0 are transport projections, an obvious inductive argument shows t* *hat the diagrams NkE _____//N0E NkE0 _____//N0E0 |g| |g| g0|| g0|| fflffl| fflffl| fflffl| fflffl| NkF _____//N0F NkF _____//N0F are pullback squares for all k 0. They are therefore homotopy cartesian by A.* *2.6 and by our assumptions. Hence it suffices to consider the case k = 0, N0u: N0E -! N0E0. By assumptions again, N0E ! N0F and N0E0 ! N0F have the concordance lifting pro* *p- erty and N0u induces a weak equivalence (homology equivalence) of the fibers. B* *y A.2.6, the fibers turn into homotopy fibers upon passage to representing spaces. Cons* *equently N0u: N0E ! N0E0 is a homotopy equivalence (homology equivalence). * * __|_| 36 In our applications of theorem 4.1.3, the categories F(X) for X in X will typi* *cally be partially ordered sets or will have been obtained from a functor Fo: C op-! sheaves on X where C is a small category. Recall that given such a functor one can define * *a category valued sheaf C opsFo on X . Its value on a manifold X is the category whose obj* *ects are pairs (c, !) with c 2 ob(C ) and ! 2 Fc(X) and whose morphisms are pairs (f, !)* * with f :b ! c in mor (C ) and ! 2 Fc(X). Then |fi(C opsFo) | ' B|C opsFo| ' hocolim |Fc| c2C (see appendix D for details). Definition 4.1.6 The sheaf fi(C opsFo): X - ! Cat will be written hocolim Fc . c2C It is in order to spell out that an element of (hocolimcFc)(X) consists of a co* *vering Y? of X indexed by the elements of J , a functor S 7! `(S) from the poset of finite n* *onempty subsets of J to C , and elements !S 2 F`(S)(YS) connected to each other via the* * maps F`(T)(YT) -! F`(S)(Y (T )) - F`(S)(Y (S)) for each S T . 4.2 Armlets We begin by defining sheaves WA and hWA on X with values in partially ordere* *d sets, and natural transformations WA___2Posets2___________________________________________* *_______________________PosetshWA22________________________________________@ ________|____________________________________|_________* *___________________________________ _________|__________________________|___________________* *________________ _____||___|__ ______||__|__ X || forget| X || forget| _____ff'|||______________________ff|'||__________________* *_________ _________|______________________________|_______________* *________________________ ________fflffl|________________________________________* *_____________________________fflffl|______________________________________@ W0 __--_________________________________SetshW0--______* *_____________________________Sets where W0 and hW0 are the sheaves introduced in section 2.3, weakly equivalent t* *o W and hW , respectively. Definition 4.2.1 An armlet for an element (ß, f) 2 W0(X) is a compact interval* * A R such that 0 2 int(A) and f is fiberwise transverse to the endpoints of A. Definition 4.2.2 An armlet for an element (ß, ^f) 2 hW0(X) is a compact interv* *al A R such that 0 2 int(A) and 37 (i) f is fiberwise transverse to the endpoints of A; (ii) ^fis integrable on an open neighborhood of f-1 (R r int(A)). We introduce a partial ordering on elements of W0(X) or hW0(X) equipped with ar* *mlets, namely for elements of W0(X): (ß, f, A) (ß0, f0,iA)f(ß, f) = (ß0, f0)and A A0 and similarly for elements of hW0(X). Definition 4.2.3 For a connected X in X we let WA (X) denote the partially or* *dered set of elements (ß, f, A) with A an armlet for (ß, f) 2 W0(X). Similarly, hWA (* *X) is the partially ordered set of elements (ß, ^f, A) where (ß, ^f) 2 hW0(X) and A and a* *rmlet for (ß, ^f). If X is not connected we (must) define Q Q WA (X) = iWA (Xi) , hWA (X) = ihWA (Xi) where the Xi are the path components of X . Any sheaf F :X ! Sets can be considered to be a sheaf with category structure, * *namely F(X) is the object set, and only identity morphisms are allowed. In this case a* *n element of fiF(X) is a pair (Y?, s) consisting of a locally finite open covering Y? = {Yj * *| j 2 J} and a single element s 2 F(X). Thus fiF ~=F x fi? where ? denotes the terminal sheaf,* * viewed as a sheaf with category values. In particular there is a forgetful projection* * fiF - ! F which is a weak equivalence. Proposition 4.2.4 The forgetful maps fiWA ! W0 and fihWA ! hW0 are weak equ* *i- valences of sheaves. The proof of proposition 4.2.4 will be broken up into the following three lemma* *s. Lemma 4.2.5 Let X be in X and (ß, f) 2 W0(X). Every x 2 X has an open neigh- borhood U in X such that the image of (ß, f) in W0(U) admits an armlet. Proof Write ß :E ! X and Ex = ß-1(x). By Sard's theorem, we can find numbers a < 0 and b > 0 such that fx: Ex ! R is transverse to a and b (in other words, * *a and b are regular values of fx). Let A = [a, b]. Let C E be the closed subset consi* *sting of all z 2 E where f has a fiberwise singularity and f(z) = a or f(z) = b. Then ß|C is* * proper and so ß(C) is a closed subset of X . Let U = X r ß(C). * * __|_| Lemma 4.2.6 With the assumptions of lemma 4.2.5, there exists an element of * *fiWA (X) mapping to (ß, f) under the forgetful transformation fiWA ! W0. 38 Proof Choose a locally finite covering of X by open subsets Yj, where j 2 J ,* * such that the restriction of (ß, f) to eachTYj admits an armlet Aj R. For a finite none* *mpty subset S J with nonempty YS let AS = j2SAj. Then AS is an armlet for the restricti* *on of (ß, f) to YS . Therefore, given nonempty finite R, S J with R S and YS 6= ;* *, we can define 'RS 2 N1WA (YS) to be the relation (ß, f, AS)|YS (ß, f, AR )|YS. If YS is empty, there is only one element in WA (YS) and so we have only one ch* *oice for 'RS . The data 'RS for all finite nonempty R, S J with R S then constitu* *te an element of fiWA (X) which clearly projects to (ß, f) 2 W(X). * * __|_| It follows from the two previous lemmas that the forgetful map fiWA [X] ! W0[X]* * is surjective for any X in X . What we really need in order to prove the first ha* *lf of pro- position 4.2.4 is the relative surjectivity as in proposition 2.2.6. This comes* * from the next lemma. Lemma 4.2.7 Let X in X and let (ß, f) 2 W0(X). Let C X be closed and supp* *ose that a germ of lifts of (ß, f) across fiWA -! W0 has been specified near C . T* *hen there exists an element in fiWA (X) which lifts (ß, f) 2 W(X) and extends the prescri* *bed germ of lifts near C . Proof Let U be a sufficiently small open neighborhood of C in X so that the p* *rescribed germ of lifts is represented by an actual lift of (ß, f)|U across fiWA (U) -! W* *0(U). This gives us a locally finite covering Y?0of U , and for each nonempty finite S J* * and each z 2 ß0(YS0), a compact interval A0S,z R such that 0 2 int(A0S,z). We have A0S,* *z A0R,~zif R S and ~zis the image of z under ß0(YS0) ! ß0(YR0). Making U smaller if nece* *ssary, we can assume that the covering Y?0is locally finite in the strong sense that e* *very x 2 X has an open neighborhood on X which intersects only finitely many of the Yj0. Now we make a locally finite covering of X by open subsets Yj as follows. For j* * 2 J such that Yj0is nonempty, let Yj = Yj0. For all other j 2 J define Yj in such a way* * that Yj avoids a fixed neighborhood of C and the restriction of (ß, f) to each path com* *ponent z 2 ß0(Yj) admits an armlet Aj,z. It remains to find enough armlets. We need one armlet AS,z R for each nonempty* * finite S J and every component z 2 ß0(YS). These armlets must satisfy AS,z AR,~z* *if R S and ~zis the image of z under ß0(YS) ! ß0(YR ). But, reasoning as in the * *proof of lemma 4.2.6, we find that it is enough to say what AS,z= Aj,zshould be when * *S is a singleton {j}. We have already said it in the cases where Yj 6= Yj0; in the oth* *er cases we say Aj,z:= A0j,z. * * __|_| The proof of the second half of proposition 4.2.4 goes like the proof of the fi* *rst half, except for an additional observation which is as follows. For X in X let hcW0(X) cons* *ist of all (ß, ^f) 2 hW0(X), with ß :E ! X etc., such that f^is integrable on some open U * * E and ß restricted to E r U is proper. 39 Lemma 4.2.8 The inclusion of sheaves hcW0 ,! hW0 is a weak equivalence. Proof Let (ß, ^f) 2 hW0(X), with ß :E ! X . Choose an open U E such that ß restricted to E r U is proper and such that the closure of U has empty intersec* *tion with f-1 (0). Using the convexity of the fibers of J2i(E, R) ! E , especially over p* *oints z 2 U , deform f^ (leaving f unchanged) in such a way that it becomes integrable on U * *. This shows that hcW0[X] ! hW0[X] is surjective. The argument can easily be refined t* *o prove a relative statement as in the hypothesis of proposition 2.2.6. * * __|_| 4.3 Proof of theorem 1.3.1 According to lemma 2.3.4 and proposition 4.2.4 it remains to show that j2i:fiWA ! fihWA is a weak equivalence. To this end we introduce a new sheaf T A :X - ! Posets. Suppose given a submersion ß :E ! X with 3-dimensional fibers and standard beha* *vior near the boundary as in condition (iii) of definitions 2.1.5 and 2.1.7. We con* *sider pairs (_, A) where _ :E ! R is a smooth function such that (ß, _): E ! X x R is prope* *r, A R is a compact interval with 0 2 int(A), and _ is fiberwise transverse to @* *A (and prescribed near @E in the usual way). There is no restriction on the fiberwise * *singularities that _ might have. Definition 4.3.1 For connected X in X , the set T A(X) consists of equivalence* * classes of triples (ß, _, A) as above, where (ß, _, A) ~ (ß0, _0, A0) if ß = ß0, A = A0* * and the support of _ - _0 is contained in the interior of _-1(A). As for WA , we get T A :X ! Posets. Moreover there is an obvious commutative di* *agram of sheaves j2ß WA =_________//=hWA" (4* *.3) == """ p==OEOE="q"""" T A where p(ß, f, A) and q(ß, ^f, A) are the equivalence classes of (ß, f, A); in t* *he second case f is the underlying function of f^. Let (ß, _, A) be a representative of an element of T A(X) with ß :E ! X , _ :E * *! R and A R. The manifold _-1(A) is independent of the choice of representative f* *or the equivalence class, and ß|_-1(A) is a proper submersion, hence a smooth fiber bu* *ndle by Ehresmann's fibration theorem [2]. Moreover, near the boundary of _-1(A), the f* *unction _ is independent of the choice of representative. 40 Lemma 4.3.2 The maps p and q in (4.3) have the concordance lifting property. Proof We give the proof for p, since the proof for q is much the same. Write * *I = [0, 1]. Given an element [ß, _, A] 2 T A(X x I) with a lift to WA (X x 0) of its restri* *ction to X x 0, the projection _-1(A) __i__//X x I (4* *.4) is a smooth manifold bundle. Hence there exists a diffeomorphism N x I ~=_-1(A)* * over X x I , where N = _-1(A) \ ß-1(X x 0). But what we need here is a diffeomorphism u: N x I -! _-1(A) over X x I such that _(u(z, t)) = _(u(0, t)) for all t 2 I and all z near @N , * *and of course u(z, 0) = z for all z 2 N . Let @h_-1(A) and @v_-1(A) be the parts of @_-1(A) which are mapped to X x (I r * *@I) and X x @I , respectively, by ß . Constructing u with the properties above is e* *quivalent to constructing a smooth vector field , = du=dt on _-1(A) which (i) covers the vector field (x, t) 7! (0, 1) 2 T Xx x T Rt on X x I , (ii)is parallel to @h_-1(A), (iii)satisfies 0 near @h_-1(A). (It should be added that , = du=dt is also prescribed near @v_-1(A) due to the * *details in definition 2.2.2.) This problem has local solutions which can be pieced togethe* *r by means of a partition of unity on _-1(A). Hence u with the required properties exists. Now we define the lifted concordance (ß, f, A) 2 WA (X xI) in such a way that f* *(u(z, t)) = f(u(z, 0)) for (z, t) 2 N x I , bearing in mind that f(u(z, 0)) is prescribed f* *or all z 2 N and f must equal _ outside u(N x I) = _-1(A). * * __|_| Proposition 4.3.3 The fiberwise jet prolongation map j2i:|fiWA | -! |fihWA | induces an isomorphism on integral homology. Proof This will be deduced from proposition 4.1.5 and diagram (4.3). By inspe* *ction, both maps p and q in (4.3) are transport projections. We must determine the fibers o* *f p and q and check that j2iinduces a homology equivalence between fibers over the same p* *oint. We first determine the fiber p-1(ø) of p: WA -! T A over an element ø = [F 3, _, A] 2 T A(?). That is, for each X in X we are inte* *rested in the subset of WA (X) which maps to the element [ß, _ O prF, A] 2 T A(X) where ß* * and 41 pr F are the projections F x X ! X and F x X ! F , respectively. This subset co* *nsists of (ß, f, A) 2 WA (X) with supp(f - _ O prF) int(_-1(A)) x X. Thus in the notation of (4.1), the fiber of p over ø is the sheaf taking X in X* * to the set of smooth maps from X to (_-1(A), A, _). Similarly, the fiber q-1(ø) of q in (4.3) over the same element ø 2 T A(?) is * *the sheaf taking X in X to the set of smooth maps from X to h (_-1(A), A, _). Thus the representing spaces |p-1(ø)| and |q-1(ø)| have canonical comparison ma* *ps to (_-1(A), A, _) and h (_-1(A), A, _), respectively, which are homotopy equivale* *nces. With these as identifications, the jet prolongation map from |p-1(ø)| to |q-1(ø* *)| turns into a special case of (4.1), and so is a homology equivalence by Vassiliev's f* *irst main the- orem. * * __|_| Combining lemma 2.3.4, proposition 4.2.4 and proposition 4.3.3, we get that j2i:|W| -! |hW| induces an isomorphism in homology. Both |W| and |hW| are topological monoids * *(cf. sections 2.3 and C.2) and j2iis a map of monoids. The target |hW| is an infinit* *e loop space by theorem 3.2.3, hence it is group complete. (That is, the monoid ß0|hW| is a* * group, or equivalently, the canonical map |hW| ! B|hW| is a homotopy equivalence). S* *ince H*(j2i; Z) is an isomorphism, especially when * = 0, the source |W| is also gro* *up complete. It is well known that a homomorphism between group complete topological monoids* * is a homology equivalence if and only if it is a homotopy equivalence. (The statemen* *t is easily reduced to the case where both monoids are connected, so that their classifying* * spaces are simply connected. One verifies that the induced map of classifying spaces is a* * homology equivalence, hence a homotopy equivalence, and deduces by applying that the o* *riginal homomorphism is a homotopy equivalence.) This completes the proof of theorem 1.* *3.1. __|_| 42 5 Some homotopy colimit decompositions 5.1 Description of main results The organization and the main results of this section can be summarized in a co* *mmutative diagram of sheaves on X and maps of sheaves WOO_______________//_WlocOO (5* *.1) |'| |'| | | LO________________//_LlocOOO | | |'| |'| hocolim LT _____//_hocolimLloc,T T inK T inK |'| |'| fflffl| fflffl| hocolim WT _____//hocolimWloc,T. T inK T inK The symbol ' indicates weak equivalences. The homotopy colimits in the diagram* * are homotopy colimits in the category of sheaves on X , as in definition 4.1.6. Bu* *t their representing spaces can be regarded as homotopy colimits in the category of spa* *ces according to lemma D.1.5. The top row of diagram (5.1) is the inclusion map W ! Wloc. The bottom row is w* *hat we eventually want to substitute for the top row in order to prove theorem 1.3.4. * *We now give a detailed description of the bottom row. This must begin with a definition of * *the category K by which the homotopy colimits are indexed. Definition 5.1.1 An object of K is a finite set S equipped with a map to 3_= * *{0, 1, 2, 3}. A morphism from S to T is a pair (k, ") where k is an injective map (over 3_) * *from S to T and " is a function T r k(S) ! {-1, +1}. The composition of two morphisms (k1, "1): S ! T and (k2, "2): T ! U is (k2k1, "3): S ! U where "3 agrees wi* *th "2 outside k2(T ) and with "1 O k2-1 on k2(T r k1(S)). Definition 5.1.2 Let T be an object of K . For X in X , let Wloc,T(X) be the s* *et of oriented, smooth, riemannian 3-dimensional vector bundles V on T x X equipped * *with a fiberwise linear isometric involution % and subject to the following conditions. (i) For (t, x) 2 T x X , the dimension of the fixed point space of -% acting o* *n the fiber V(t,x)is equal to the label of t in 3_; 43 (ii) The composition V ! T x X ! X is a graphic map. A smooth map g :X ! Y induces a map Wloc,T(Y ) ! Wloc,T(X), given by pullback * *of vector bundles along idx g :T x X ! T x Y . This makes Wloc,Tinto a sheaf on X . In definition 5.1.2, the involution on V leads to an orthogonal vector bundle * *splitting V = V % V -%, where V %consists of the vectors fixed by % and V -% consists of* * the vectors fixed by -%. In the next definition, D(V %) and S(V -%) denote the disk and sph* *ere bundles associated with V %and V -%, respectively. Definition 5.1.3 For T in K , a sheaf WT on X is defined as follows. For X in* * X , an element of WT(X) consists of (i) a smooth graphic bundle q :M ! X of compact oriented surfaces; (ii) an element (V, %) of Wloc,T(X); (iii) a smooth and fiberwise orientation preserving embedding over X , e: D(V %) xTxX S(V -%) -! M r @M . Boundary condition: Near their respective boundaries, the manifolds M and (S1x[* *0, 1])xX agree, and there q agrees as an oriented map with the projection to X . The sheaves defined in 5.1.2 and 5.1.3 depend contravariantly on the variable T* * in K . This is clear in the case of 5.1.2: A morphism (k, "): S ! T in K induces a m* *ap from Wloc,T(X) to Wloc,S(X) given by pullback of vector bundles along k x id:S x X !* * T x X . The case 5.1.3 is much more interesting. Let (k, "): S ! T be a morphism in K* * . If k is bijective, there is an obvious identification WT ~= WS and this is the ind* *uced map. Therefore we may assume that k is an inclusion S ,! T . Then we can reduce to t* *he case where T r S has exactly one element, a. This case has two subcases: ä( ) = +1* * and ä( ) = -1. Definition 5.1.4 Let (k, "): S ! T be a morphism in K where k is an inclusio* *n and T r S = {a} with ä( ) = +1. We describe the induced map WT(X) -! WS(X). Let (q, V, %, e) be an element of WT(X), with q :M ! X . Map this to an element* * of WS by keeping q :M ! X , restricting V to S x X and restricting % and e according* *ly. Definition 5.1.5 Let (k, "): S ! T be a morphism in K where k is an inclusio* *n and T r S = {a} with ä( ) = -1. For X in X , the induced map WT(X) -! WS(X) is defined as follows. Let (q, V, %, e) be an element of WT(X), with q :M ! X .* * Map this to the element (q0, V 0, %0, e0) of WS where 44 (1) q0: M0 ! X is the surface bundle obtained from q :M ! X by fiberwise surg* *ery on the embedded bundle of thickened spheres e D(V %|Xa) xXa S(V -%|Xa) , whe* *re Xa means a x X ; (2) (V 0, %0) is the restriction of (V, %) to S x X ; (3) e0 is obtained from e by restriction. Remark 5.1.6 The fiberwise surgery in (i) amounts to removing the interior o* *f the em- bedded thickened sphere bundle and gluing in a copy of D(V -%|Xa)xXa S(V %|Xa) * *instead. Note in particular that when V -% = 0, the embedded thickened sphere bundle who* *se in- terior we have to remove is empty. In this case the fiberwise surgery consist * *in adding a disjoint copy of the sphere bundle S(V )|Xa to M . Remark 5.1.7 To ensure that T 7! WT really is a contravariant functor on K * *, it is wise to add two conditions of a set-theoretic nature to definition 5.1.3. Namel* *y, e should be an inclusion and M r im(e) should have no elements in common with V . Surger* *ies as in definition 5.1.5 should then be performed by removing something from M which* * also happens to be a subset of V , and gluing in another subset of V . (All thicken* *ed sphere bundles in sight can be thought of as subbundles of V .) There is a forgetful map of sheaves WT ! Wloc,T. It has the concordance lifting* * property, so that the representing spaces of its fibers are the homotopy fibers of the in* *duced map or representing spaces |WT| ! |Wloc,T|. It is easy to see that the representing space of any fiber of WT ! Wloc,Tis a c* *lassifying space for certain bundles of compact oriented surfaces with fixed boundary. 5.2 Morse singularities, Hessians and surgeries We begin by recalling some well known facts about elementary and multi-elementa* *ry Morse functions. The reader is referred to [27, ch.I] and [28] for more details in th* *e non-parame- trized situation. By an elementary Morse function we shall mean a proper smoot* *h map E ! R which is regular on @E and has exactly one critical point in E r @E , tha* *t one nondegenerate. By a multi-elementary Morse function we mean a proper smooth map E ! R which is regular on @E and has finitely many critical points in E r @E ,* * all nondegenerate and all with the same critical value. Fix a finite dimensional real vector space V with an inner product (i.e., a po* *sitive definite bilinear form) and a linear isometric involution %: V ! V . Then the function f* *V :V ! R given by fV (v) = (5* *.2) 45 is a Morse function on V with exactly one critical point. If we write V = V % * * V -%, then the fomula for fV becomes fV (v) = kv+ k2 - kv- k2 where v+ and v- are the components of v in V % and V -%, respectively. The g* *radient of fV on V is everywhere perpendicular to the gradient of v 7! kv+ k2kv- k2, * *so that the latter function is constant on the trajectories of the gradient flow of fV . T* *his motivates the following definition. fi Definition 5.2.1 saddle(V, %) = {v 2 V fikv+ k2kv- k2 1}. If V %= 0 or V -%= 0, then saddle(V, %) = V . In the remaining cases, the formu* *la v 7! (kv- kv+ , kv- k-1v- , ) (5* *.3) defines a smooth embedding of saddle(V, %)rV %in D(V %)xS(V -%)xR, with complem* *ent 0 x S(V -%) x [0, 1[. It respects boundaries and takes the gradient vectors of * *v 7! to tangent vectors which are parallel to the R factor. It is a map over R, wher* *e we use the restriction of fV on the source and the function (x, y, t) 7! t on the target. Dually, the formula v 7! (kv+ kv- , kv+ k-1v+ , ) (5* *.4) defines a smooth embedding of saddle(V, %)rV -% in D(V -%)xS(V %)xR, with compl* *ement 0 x S(V %)x ] - 1, 0]. It respects boundaries and takes the gradient vectors of* * v 7! to tangent vectors which are parallel to the R factor. It is also a map over R. The map fV in (5.2) restricted to saddle(V, %) is a good local model for elemen* *tary Morse functions. Let M be any smooth compact manifold and let e: D(V %) x S(V -%) ! M r @M (5* *.5) be a codimension zero embedding (üs rgery data"), assuming dim(V ) = dim(M)+1. * *Then in M x R we have an embedded copy of D(V %) x S(V -%) x R. We can remove its in* *terior and glue in saddle(V, %) instead, using formula (5.3) to identify the boundary * *of saddle(V, %) with the boundary of D(V %) x S(V -%) x R. The result is a smooth manifold trac* *e(e) of dimension dim (M) + 1. More precisely: Definition 5.2.2 The long trace of e, denoted trace(e), is the pushout of the * *two smooth codimension zero embeddings (exid)O(5.3) saddle(V, %) r V %_____________//(M x R) r e(0 x S(V -%)) x [0, 1[ , (5* *.6) " saddle(V, %) r VØ%_____________//saddle(V, %). 46 For example, if V -% = 0, then saddle(V, %) = V and saddle(V, %) r V % is empt* *y, so that trace(e) becomes the disjoint union of M x V and V = V %. If V %= 0, then M co* *ntains a codimension zero copy of S(V ). The long trace is obtained by removing S(V ) * *x [0, 1[ from the copy of S(V ) x R in M x R and adding a single point instead, so that * *trace(e) becomes the disjoint union of (M r im(e)) x R and V = V -%. The description 5.2.2 determines a structure of smooth manifold on trace(e) and* * shows that trace(e) comes with a (smooth) elementary Morse function, the height function, * *which is the projection to R on the complement of V %and equal to v 7! on the gl* *ued-in copy of saddle(V, %). The unique critical point is the origin of V % trace(e). The * *corresponding critical value is 0. Conversely, suppose that N is any smooth manifold with boundary and g :N ! R is* * an elementary Morse function, with critical value 0 and unique critical point z 2 * *N r@N . Let V = T Nz. Choose an exponential map h: V ! N r @N , an inner product < , > on V* * , a linear isometric involution % on V and ffi > 0 such that gh(v) = for a* *ll v 2 V with < ffi . This is possible by the Morse-Palais lemma; see for example [23]* *. At the price of replacing g by 3ffi-1g , we can assume ffi = 3. Now choose a smooth vector * *field , on N which extends h*(grad(gh)) on h(D(V %) x D(V -%)), is tangential to @N and sa* *tisfies > 0 on N r z . For the function f :V ! R given by v 7! , we the* *n have a unique smooth embedding ~h:saddle(V, %) ! N which extends h on D(V %) x D(V -* *%), maps gradient flow trajectories of f to flow trajectories of , and satisfies g* * O ~h= f on saddle(V, %). This identifies N with a long trace. The long trace construction has some obvious generalizations. For example, we c* *an allow simultaneous surgeries on a finite number of pairwise disjoint thickened sphere* *s. In this case the surgery data consist of a finite set T , a riemannian vector bundle V * * on T with an isometric involution %, where dim (V ) = dim(M) + 1, and a smooth embedding e: D(V %) xT S(V %) -! M r @M . Then trace(e) is defined as the manifold obtained from M x R by deleting the em* *bedded copy of D(Vt%) x S(Vt-%) x R for each t 2 T , and substituting saddle(Vt, %) for it using formula (5.3) to d* *o the gluing. There is a canonical height function on trace(e). It is a Morse function with o* *ne critical point for each t 2 T . The only critical value is 0 (if T 6= ;). Then there is a parametrized version of the previous construction. Let q :M ! * *X be a bundle of smooth compact n-manifolds, let V ! T x X be a riemannian vector bund* *le of fiber dimension n + 1 with isometric involution %, and let e: D(V %) xTxX S(V -%) -! M r @M 47 be a smooth embeding over X . We can regard e as a family of embeddings ex for * *x 2 X , each from a disjoint union of finitely many thickened spheres to a fiber Mx of * *q . The manifolds trace(ex) for x 2 X are the fibers of a smooth bundle trace(e) -! X . (5* *.7) It comes equipped with a smooth height function f :trace(e) -! R which is fiber* *wise Morse; if T 6= ;, then the unique critical value is 0. For a useful naturality property of saddle(V, %), suppose given a smooth orient* *ation pre- serving embedding e: R ! R such that e(0) = 0. Let fV : V ! R be the canonical quadratic function, fV (v) = . Proposition 5.2.3 There is a diffeomorphism ~: saddle(V, %)-! f-1V(e(R)) \ saddle(V, %) such that fV ~ = efV on saddle(V, %) and ~0(0) = e0(0) x idV. Proof The Morse-Palais lemma as presented in [23], for example, gives us the * *germ of ~ near 0 2 saddle(V, %) V . We may assume that this is defined on a neighborho* *od of a subset of saddle(V, %) of the form fi Ka,b= {v 2 V fikv+ k2kv- k2 a and |fV (v)| b} where a and b are small positive numbers (to begin with). The boundary @Ka,bis* * the union of a vertical and a horizontal part, fi @0Ka,b = {v 2 V fikv+ k2kv- k2 = a and |fV (v)| b}, fi @1Ka,b = {v 2 V fikv+ k2kv- k2 a and |fV (v)| = b}. Since fV and efV are both regular away from 0, in particular outside Ka,b, it* * is easy to construct an extension to all of saddle(V, %) having the required properties. T* *he extension can be made in two steps. In the first step, note that K1,bis the union of Ka,b* *and a closed collar on @0Ka,b. Use this to define ~ on all of K1,b, in such a way that K1,bi* *s mapped diffeomorphically to fi {v 2 saddle(V, %) fi|e-1fV (v)| b}. Then note that saddle(V, %) = K1,1 is the union of K1,band an open collar on @1* *K1,b. Use this to define ~ on all of saddle(V, %). * * __|_| 5.3 Second row As the subsection title indicates, we are going to concentrate on the second ro* *w of dia- gram (5.1), but we will also explain how it is related to the first row. 48 Definition 5.3.1 Let Llocbe the following sheaf on X . For X in X , an elemen* *t of Lloc(X) is a triple (p, g, V ) where (i) p is a graphic and 'etale map from some smooth Y to X ; (ii) g is a smooth function Y ! R ; (iii) V is an oriented, smooth, riemannian 3-dimensional vector bundle on Y e* *quipped with a linear isometric involution %: V ! V over Y . Conditions: The map (p, g): Y ! X x R is proper and the composition of the ve* *ctor bundle projection V ! Y with p: Y ! X is a graphic map. Remark 5.3.2 Let (ß, f) be an element of Wloc(X), with ß :E ! X and f :X ! R. Let = (ß, f) be the fiberwise singularity set of f . We showed in lemma 2.1* *.3 that ß| is 'etale. By definition of Wloc(X), the map (ß| , f| ): ! X x R is pro* *per. The restriction V of the vertical tangent bundle T iE to comes with an every* *where nondegenerate symmetric bilinear form 1_2H , where H is the vertical Hessian o* *f f . See [27, I,x2]. We can choose an orthogonal splitting V = V + V - of T iE| into a* * positive definite subbundle and a negative definite subbundle. By changing the sign of 1* *_2H on V - , we make V into a riemannian vector bundle, with an isometric involution which * *is -id on V - and +id on V +. In this way, the element (ß, f) of Wloc(X) determines an el* *ement (ß| , f| , V ) 2 Lloc(X). We come to the construction of the map from Llocto Wlocwhich appears in diagram* * (5.1). Fix X in X and let (p, g, V ) be an element of Lloc(X), with p: Y ! X and isom* *etric involution %: V ! V . Let E be the disjoint union of V and (S1 x R x [0, 1] ) * *x X . Let ß :E ! R agree with the composition V ! Y ! X on the summand V , and with the projection to X on the other summand. Let f :E ! R be given by f(v) = g(y) + for y 2 Y and v in the fiber of V over y , and let f agree with the projectio* *n to R on the other summand. Then (ß, f) is an element of Wloc(X). The rule (p, g, V ) 7! (ß,* * f) is our map. Proposition 5.3.3 The map Lloc! Wlocso defined is a weak homotopy equivalence. Proof We are going to use the relative surjectivity criterion of proposition * *2.2.6. To deal with the absolute case first, we assume given X in X and (ß, f) 2 Wloc(X)* *, with ß :E ! R and f :E ! R. Let = (ß, f) be the fiberwise singularity set of f . Choose a tubular neighborhood V of in E such that the vector bundle projec* *tion V ! is over X . This is possible by lemma 2.1.3. Choose an open neighborho* *od E0 of @E in E which, as a space over X x R, agrees with an open neighborhood of * *the boundary in (S1 x R x [0, 1]) x X . This is possible because of the boundary co* *ndition in the definition of Wloc(which is identical with the boundary condition in defini* *tion 2.1.6). 49 By proposition 3.4.1, the element (ß, f) in Wloc(X) is concordant to (ß(1), f(1* *)) where ß(1) and f(1) are the restrictions of ß and f to V [ E0, respectively. Another appli* *cation of proposition 3.4.1 gives us that (ß(1), f(1)) is concordant to (ß(2), f(2)), whe* *re ß(2)and f(2) are the canonical extensions of ß(1)and f(1)to V q (S1 x R x [0, 1]) x X . In particular f(2) is still equal to f on V and is the projection to R on the* * summand (S1 x R x [0, 1]) x X . The next step is to improve f(2)|V = f|V . Let _ :[2, 3] ! [0, 1] be a smooth non-increasing function such that _(t) = 1 f* *or t close to 2 and _(t) = 0 for t close to 3. For t 2 [2, 3] let f(t)be given by æ -2 v 7! fp(v) +f_(t)p((f(_(t)v)v-)fp(v))+ for1_(t)_> 0 and v 2 V 2H(pv)(v,fv)or _(t) = 0 and * *v 2 V where H(pv) denotes the vertical Hessian of f at p(v), alias second derivative* * in the fiber direction. Let f(t)agree with f(2) on (S1 x R x [0, 1]) x X and let ß(t)=* * ß(2)for convenience. Then t 7! (ß(t), f(t)) defines a concordance, parametrized by the* * interval [2, 3], from (ß(2), f(2)) to (ß(3), f(3)). To lift (ß(3), f(3)) to an element o* *f Lloc(X) we only need to choose a maximal negative definite subbundle of V for (half) the verti* *cal Hessian. Compare remark 5.3.2. We have now established the absolute case of the relative* * surjectivity condition of 2.2.6 for our map Lloc! Wloc. The relative case is not much more d* *ifficult and we leave it to the reader. * * __|_| For later use we note the following: Lemma 5.3.4 Let (p, g, V ) 2 Lloc(X), with p: Y ! X . For every x 2 X and * *every b > 0 there exist a neighborhood U of x in X such that, on every component of p* *-1(U), the function g is either bounded below by -b or bounded above by b. Proof Chose a descending sequence of open balls Ui for i = 0, 1, 2, 3, . .f.o* *rming a neigh- borhood basis for x in X . If the statement is false, then there exists b > 0 a* *nd connected subsets Ki Y for i = 0, 1, 2, 3, . .s.uch that p(Ki) Ui and g(Ki) [-b, b* *] for all i. Choose zi 2 Ki such that g(zi) = 0. The sequence z0, z1, z2, . .i.n Y mus* *t have a convergent (infinite) subsequence, because (p, g): Y ! X x R is proper and the * *two image sequences in X and R converge. Let z1 2 Y be the point which the subsequence co* *nverges to. Then p(z1 ) = x and g(z1 ) = 0. Now p: Y ! X is 'etale. Hence, for sufficie* *ntly large i, there are unique neighborhoods U0iof z1 in Y such that p maps U0idiffeomorp* *hically to Ui. It follows that zi2 U0ifor infinitely many i and hence Ki U0ifor infinitel* *y many i. But it is also clear that the diameter of g(U0i) tends to zero as i tends to in* *finity; hence the lim inf of the diameters of the intervals g(Ki) is zero, which contradicts our * *assumption. __|_| We turn to the left hand side of the second row in diagram (5.1). Definition 5.3.5 For X in X , an element of L(X) shall consist of 50 (i) an element (ß, f) 2 W(X), with ß :E ! X etc., (ii) an element (p, g, V ) 2 Lloc(X), with p: Y ! X etc., (iii) a smooth, orientation preserving embedding ~: saddle(V, %) -! E r @E over* * X x R such that im(~) contains (ß, f). fi Here saddle(V, %) is a subbundle of V , defined as {v 2 V fikv+ k2kv- k2 1}. * * We have a canonical map V ! R defined by v 7! g(y) + for v in a fiber Vy of the* * vector bundle V . (This was also used in the construction of the map Lloc! Wloc.) In* * this way, saddle(V, %) becomes a space over X x R. The conditions in (iii) then impl* *y that ~ identifies the zero section of V with the fiberwise singularity set (ß, f). Remark 5.3.6 The embedding ~: saddle(V ) ! E r @E need not have a closed ima* *ge, because the 'etale map Y ! X need not be a closed map. But im(~) is locally com* *pact, therefore locally closed in E . Proposition 5.3.7 The forgetful map L ! W is a weak homotopy equivalence. Proof Again we use the relative surjectivity criterion of proposition 2.2.6 a* *nd again we begin with the absolute case. Fix X in X and (ß, f) 2 W(X), with ß :E ! X . We* * want to lift the concordance class of (ß, f) to a class in L[X]. As in the proof of * *proposition 5.3.3, we begin by choosing a vertical tubular neighborhood V E of = (ß, f), with* * vector bundle projection ! :V ! over X . Then V is canonically identified with T iE| , the restriction of the * *vertical tangent bundle of E to . Using this identification, we define fV :V ! R by fV (v) = 1_2H(!(v))(v, v) , where H(!(v)) is the second derivative of f , at !(v), in the vertical directio* *n. (This second derivative is a symmetric bilinear form on the vertical tangent space at* * !(v), well defined because the vertical part of the first derivative of f at !(v) vanishe* *s.) By the Morse-Palais lemma [23], we can choose the vector bundle structure on V in suc* *h a way that f(v) = fV (v) + f!(v) holds in a neighborhood U of the zero section of V .* * Next we choose a positive definite inner product on V such that fV (v) = for a * *(unique) linear isometric involution %: V ! V . Without loss of generality, the neighborhood U * *contains all v 2 saddle(V, %) for which |f!(v)| 1 and |fV (v) + f!(v)| 1. If not, re* *place f by _f where _ :E ! [1, 1[ is a suitable smooth function which factors through ß :E* * ! X . Multiply the inner product on V by _ , too. The pairs (ß, f) and (ß, _f) are c* *oncordant. Summarizing, we can arrange f(v) = fV (v) + f!(v) for v 2 V with |f!(v)| 1 and |fV (v) + f!* *(v)|. 1 51 Now choose a smooth embedding e: R ! R with im(e) = ] - 1, 1[ , equal to the id* *entity near 0 2 R. Then (ß, f) is concordant to (ß], f]), where ß] is the restriction* * of ß to E] = f-1 (im (e)) and f] is e-1f on E]. Let ] = \ E] and V ]= V | ]. Let n fi o K = v 2 saddle(V ], %) fi|fV (v) + f!(v)|.< 1 Then f|K = fV |K + f!|K by our assumptions, so K E]. Using proposition 5.2.3,* * we can construct an orientation preserving diffeomorphism ~: saddle(V ], %) -! K relative to and over ], such that (fV + f!)~ = e(fV + f!). This can also be vi* *ewed as an embedding of saddle(V ], %) in E]. Then f]~ = e-1f~ = e-1(fV + f!)~ = e-1e(fV + f!) = fV + f! on saddle(V ], %). That is, ~ promotes the pair (ß], f]) to an element of L(X)* *. This establishes the absolute case of the relative surjectivity condition. The truly relative case is slightly more difficult. We sketch it. Again fix X* * in X and (ß, f) 2 W(X), with ß :E ! X . Let C X be closed. We want to find an element in L(X) whose image in W(X) is concordant rel C to (ß, f). This can be constru* *cted essentially as in the absolute case, except for one change which consists in re* *placing the embedding e: R ! R above by a smooth family of smooth embeddings ex: R ! R, depending on x 2 X . Then we have the option to choose ex = idR for x in a very small neighborhood of C , while having im (ex) [-1, +1] for x outside a sligh* *tly larger neighborhood of C . * * __|_| 5.4 Third row Definition 5.4.1 Fix S in K and X in X . We define a sheaf Lloc,Son X . For* * X in X , an element of Lloc,S(X) is an element (p, g, V ) of Lloc(X), where p has* * source Y , together with (i) an embedding h: S x X ! Y over 3_x X , (ii) a continuous function ffi :Y r im(h) -! {-1, +1}. Condition: Every x 2 X has a neighborhood U in X such that g admits a lower bou* *nd on p-1(U) \ ffi-1(+1) and an upper bound on p-1(U) \ ffi-1(-1). In definition 5.4.1, the function ffi has to be constant on each component of Y* * r im(h). Using partitions of unity, one can reformulate the local bound condition by a g* *lobal one, as follows: there exists a continuous function b: X ! R such that -bp g on ffi-1* *(+1) and bp g on ffi-1(-1). 52 A morphism (k, "): R ! S in K induces a map Lloc,S! Lloc,Rtaking an element (p, g, V, h, ffi) of Lloc,S(X) to (p, g, V 0, h0, ffi0) where V 0is obtained fr* *om V by pulling back, h0(r, x) = h(k(r), x) for (r, x) 2 R x X and æ ffi0(y) = ffi(y)if"ffi(y)(issdefined)if(ys=,h(s,xx))for2some(S r * *k(R)) x X. This makes the rule T 7! Lloc,Tinto a contravariant functor from K to the cate* *gory of sheaves on X . Moreover, for each T in K there is a forgetful map Lloc,T! Lloc. The maps Lloc,T! Lloc,Sinduced by morphisms S ! T in K are over Lloc. This lea* *ds to a canonical map of sheaves v : hocolim Lloc,T - ! Lloc. (5* *.8) T inK Proposition 5.4.2 The map v in (5.8) is a weak equivalence. Proof Let Lffilocbe the following sheaf on X with category structure. An obje* *ct of Lffiloc(X) is an element (p, g, V ) of Lloc(X), with p: Y ! X , together with a continuous* * function ffi :Y ! {-1, 0, +1} subject to the following condition: Every x 2 X has a neighborhood U in X such that g admits a lower bound on p-1(U) \ ffi-1(+1), an upper bound on p-1(U) \ ffi-1(-1), and both an upp* *er and a lower bound on p-1(U) \ ffi-1(0). Given two such objects, (p, g, V, ffia) and (p, g, V, ffib) with the same under* *lying (p, g, V ), we write (p, g, V, ffia) (p, g, V, ffib) if ffi-1a(+1) ffi-1b(+1) and ffi-1a(-* *1) ffi-1b(-1). In this situation there is a unique morphism from (p, g, V, ffia) to (p, g, V, ffib), o* *therwise there is none. Thus the category Lffiloc(X) is a poset. The map v in (5.8) can now be factorized as follows: hocolim Lloc,T_v1_//_fiLffilocv2//_Lloc (5* *.9) T inK Here v2 is induced by the forgetful map Lffiloc! Lloc. (Compare proposition 4.* *2.4.) To describe v1 we recall that hocolimT Lloc,Twas defined as fi(K opsLloc,o). An object in (K opsLloc,o)(X) consists of an object T in K and an element a in* * Lloc,T(X). A morphism from (T, a) to (S, b) is a morphism S ! T in K taking a to b. An ob* *ject (T, a) in (K opsLloc,o)(X) with a = (p, g, V, ffi, h) determines an object (p, g, V, ~ffi) 53 in Lffiloc(X), where ~ffi(z) = ffi(z) if ffi(z) is defined and ~ffi(z) = 0 othe* *rwise. This canonical association is a functor, for each X , and as such induces v1. For simply connected X , the functor so defined, from (K opsLloc,o)(X) to Lffil* *oc(X), is clearly an equivalence of categories. (The point is that, by Ehresmann's fibrat* *ion theorem, a proper 'etale map to a simply connected manifold is always a trivial bundle w* *ith finite fiber.) In particular, it is an equivalence of categories for the extended simp* *lices, X = ke where k 0. Consequently v1 in (5.9) is a weak equivalence. Compare section 4.* *1. With a view to showing that v2 is also a weak equivalence, we make the followin* *g obser- vation. Given objects (p, g, V, ffi1) and (p, g, V, ffi2) in Lffiloc(X), with * *the same underlying (p, g, V ) 2 Lloc(X), there always exists an object (p, g, V, ffi3) in Lffiloc(* *X) such that (p, g, V, ffi3)(p, g, V, ffi1) (p, g, V, ffi3)(p, g, V, ffi2). Namely, let ffi3(z) = +1 if and only if ffi1(z) = +1 = ffi2(z); let ffi3(z) = -* *1 if and only if ffi1(z) = -1 = ffi2(z), and let ffi3(z) = 0 in the remaining cases. Now we apply proposition 2.2.6 to v2. Given (p, g, V ) 2 Lloc(X), we can by lem* *ma 5.3.4 find a locally finite covering of X by open subsets Uj, where j 2 J , such that* * (p, g, V ) |U has a lift 'jj to ob(Lffiloc)(Uj) for all j . With the observation just above, * *it is easy to extend the collection of the 'jj to a collection of elements 'RR 2 ob(Lffiloc)(UR ), * *in such a way that 'RR 'QQ |UR whenever Q R. The collection of these 'RR is then an elem* *ent of fiLffiloc(X). This establishes the absolute case of the hypothesis in 2.2.6, an* *d the verification is much the same in the relative case. * * __|_| Definition 5.4.3 For T in K , we define a sheaf LT as the pullback of forget forget L _________//_Llocoo______Lloc,T. Remark 5.4.4 An element in LT(X) consists of (ß, f) 2 W(X) with ß :E ! X , an element (p, g, V, h, ffi) in Lloc,T(X), with p: Y ! X and h: T x X ! Y , and a * *smooth embedding ~: saddle(V, %) ! E over X x R satisfying condition (iii) in definiti* *on 5.3.5. Here E denotes the source of ß and f . Definition 5.4.3 leads to a canonical map u from hocolimT LT to L. Proposition 5.4.5 The map u: hocolim LT -! L is a weak equivalence. T inK Proof The proof is completely analogous to that of proposition 5.4.2. There i* *s a factoriz- ation of u having the form hocolim LT __u1_//fiLffiu2//_L (5.* *10) T inK where Lffiis defined as the pullback of L -! Lloc- Lffiloc. One shows that u1 * *and u2 are weak equivalences. * * __|_| 54 5.5 Fourth row, right hand column Definition 5.5.1 Fix T in K . We define a map from Lloc,Tto Wloc,Tby Lloc,T(X) 3 (p, g, V, h, ffi) 7! h*(V ) 2 Wloc,T(X). To make sense of this formula, recall that p: Y ! X denotes an 'etale map, V d* *enotes a riemannian vector bundle with involution on Y and h: T x X ! Y is an embeddin* *g over X . Therefore h*(V ) is a riemannian vector bundle with involution on T x X . There is an equally simple map in the other direction, Wloc,T! Lloc,T. Namely,* * for X in X we can identify Wloc,T(X) with a subset of Lloc,T(X), consisting of the e* *lements (p, g, V, h, ffi) 2 Lloc,T(X) which have h = idTxX and g 0. Lemma 5.5.2 The inclusion Wloc,T! Lloc,Tis a weak equivalence. Proof We use proposition 2.2.6. Given (p, g, V, h, ffi) 2 Lloc,T(X) with p: Y* * ! X , choose a smooth _ :[0, 1=2[ ! [0, 1[ such that _(s) = 0 for s close to 0 and _(s) tend* *s to +1 for s ! 1=2. Choose another smooth ': [0, 1] ! [0, 1] such that '(s) = 1 for s * *close to 0 and '(s) = 0 for s close to 1. Then define (~p, ~g, ~V, ~h, ~ffi) 2 Lloc,T(Xx ]0, 1[ ) in the following way. The source of ~pis the union of Y x ]0, 1=2[ and h(T x * *X)x ]0, 1[ . The formula for ~pis ~p(y, s) = (p(y), s). (To ensure that ~pis graphic, we sh* *ould define the source of p~ and ~g as a subset of the pullback of p: Y ! X along the pro* *jection Xx ]0, 1 [ -! X . See definition 2.1.1.) The formula for g~ is g~(y, s) := g(* *y) . '(s) if y is in h(T x X) and g~(y, s) := g(y) + ffi(y)_(s) otherwise. The vector bund* *le V~ is the pullback of V under the projection. The formula for ~h is ~h(t, x, s) := * *(h(t, x), s) and the formula for ~ffiis ~ffi(y, s) = ffi(y). By inspection, (~p, ~g, ~V, ~h* *, ~ffi) is a concordance from (p, g, V, h, ffi) 2 Lloc,T(X) to an element (p[, g[, V [, h[, ffi[) 2 Lloc* *,T(X) where h[ is a homeomorphism and g[ 0. With some renaming we can arrange h[ to be an identi* *ty map, so that (p[, g[, V [, h[, ffi[) 2 Wloc,T(X). If a closed subset C of X is * *given, and the restriction of (p, g, V, h, ffi) to some open neighborhood U of C is already i* *n Wloc,T(U), then the concordance just constructed is constant on U , giving the relative su* *rjectivity condition in proposition 2.2.6. * * __|_| Corollary 5.5.3 The map Lloc,T! Wloc,Tof definition 5.5.1 is a weak equivalenc* *e. Proof The composite map, from Wloc,Tto Lloc,Tand back to Wloc,T, is clearly a* * weak equivalence. * * __|_| 55 5.6 Fourth row, left hand column The goal is to write down a map LT ! WT , depending naturally on T in K , and t* *o show that it is a weak equivalence. We organise the information contained in a single element of LT(X) as in remark* * 5.4.4 and use the same notation. Write ! :V ! Y for the vector bundle projection, < , * *> for the inner product on V and %: V ! V for the isometric involution, as usual. In ad* *dition let C+ = {v 2 saddle(V, %) | ffi(!(v)) = +1 and -1}, C0 = {v 2 saddle(V, %) | ffi(!(v)) undefined}, C- = {v 2 saddle(V, %) | ffi(!(v)) = -1 and +1}. The images ~(C+ ), ~(C0) and ~(C- ) are closed subsets of E , despite remark 5.* *3.6. They are also codimension zero submanifolds of E , with corners in the case of C+ a* *nd C- . Writing U for the various connected components of V , let a Crg+ = D(U%) xY S(U-% ) x [-1, 1[ , U withffi!|U +1 a Crg0 = D(U%) xY S(U-% ) x R , U withffi!|U undef. a Crg- = D(U-% ) xY S(U%)x ] - 1, +1] . U withffi!|U -1 Then we have identifications @C+ ~= @Crg+ by (5.3), @C0 ~= @Crg0 by (5.3), @C- ~= @Crg- by (5.4). Definition 5.6.1 The regularization Erg of E above is obtained by removing the* * interior of the closed codimension zero submanifold ~(C+ [ C0 [ C- ) and gluing in a cop* *y of Crg+[ Crg0[ Crg-instead, using the boundary identifications @C+ ~=@Crg+, @C0 ~=* *@Crg0and @C- ~=@Crg-just defined. Remark 5.6.2 More precisely, Erg is defined in two steps. First, remove ~(C* *+ \ V %), ~(C0 \ V %) and ~(C- \ V -%) from E . The result is a manifold with disjoint, * *properly embedded codimension zero copies of C+ r V %, C0 r V % and C- r V -%. Then mak* *e a (triple) cobase change along the codimension zero embeddings C+ r V % -! Crg+, C0 r V % -! Crg0, C- r V -% -! Crg- 56 determined by (5.3) and (5.4). This description gives a preferred structure of* * smooth manifold on Erg. The manifolds Crg+, Crg0and Crg-come with a canonical map to X , via the projec* *tions to Y . They also come with a canonical map to R, given by (v, t) 7! t + . U* *nder the identifications in 5.6.2, these maps match ß :E ! X and f :E ! R, respectively,* * which leads to well defined and smooth maps ßrg: Erg ! X, frg: Erg ! R. By construction, ßrg is still a submersion and the product map (ßrg, frg): Erg * *! X x R is still proper. But in addition frg is regular when restricted to any fiber of* * ßrg. Therefore and by Ehresmann's fibration theorem we have proved Proposition 5.6.3 The map (ßrg, frg): Erg ! X x R is a bundle of smooth compa* *ct surfaces. * * __|_| Keeping the above notation, let M = {z 2 Erg | frg(z) = 0} and let q = ßrg|M . * * Then q :M ! X is a bundle of smooth compact surfaces by proposition 5.6.3. The inter* *section of M with the embedded copy of Crg0in Erg is identified with D(h*V %) xTxX S(h*V -%). (5.* *11) The surface bundle q :M ! X , the riemannian vector bundle h*V ! T x X and the canonical embedding e of (5.11) in M now constitute an element of WT(X). Definition 5.6.4 The map LT ! WT promised in diagram (5.1) takes the element * *in remark 5.4.4 to (q, h*V, e) 2 WT(X), where q = ßrg|M and M = {z 2 Erg| frg(z) =* * 0}. By inspection, the map 5.6.4 is natural in the variable X and in the variable T* * , where T runs through the objects of K . It makes diagram (5.1) commutative. Remark 5.6.5 There is a slightly different way to describe the regularizatio* *n process; it will help us in proving that the map LT ! WT defined in 5.6.4 is a weak equiva* *lence. Keeping the notation of remark 5.4.4, let K+ = {v 2 saddle(V, %) | ffi(!(v)) = +1}, K0 = C0 = {v 2 saddle(V, %) | ffi(!(v)) undefined}, K- = {v 2 saddle(V, %) | ffi(!(v)) = -1}, 57 so that saddle(V, %) = K+ [ K0 [ K- . The regularized versions are a Krg+ = D(U%) xY S(U-% ) x R , U withffi!|U +1 a Krg0 = D(U%) xY S(U-% ) x R , U withffi!|U undef. a Krg- = D(U-% ) xY S(U%) x R. U withffi!|U -1 We can re-define Erg as follows: First, remove ~(K+ \ V %), ~(K0\ V %) and ~(K-* * \ V -%) from E . The result is a manifold with disjointly embedded codimension zero co* *pies of K+ rV %, K0rV %and K- rV -%. Then make a (triple) cobase change along the codim* *ension zero embeddings K+ r V % -! Krg+, K0 r V % -! Krg0, K- r V -% -! Krg- determined by (5.3) and (5.4). Comparison with remark 5.6.2 shows that this new description of Erg agrees with* * the old one up to a canonical diffeomorphism (over X xR). The new description has the a* *dvantage of giving us a canonical (in general non-closed) codimension zero embedding Krg+[ Krg0[ Krg--! Erg. Intersecting its image with M = {z 2 Erg| frg(z) = 0}, we get a canonical (and * *in general non-closed) embedding D(V ~%) xY S(V -~%) -! M (5.* *12) where ~%:V ! V is equal to -% on connected components of V having ffi! -1,* * and ~% = % elsewhere on V . This extends the embedding of (5.11) into M and leads* * to a factorization of the map LT ! WT from definition 5.6.4, which we now make expli* *cit. Definition 5.6.6 Let W^T be the sheaf on X defined as follows. For X in X , a* *n element of W^T(X) consists of (i) a smooth graphic bundle q :M ! X of compact oriented surfaces; (ii) an element (p, g, V, h, ffi) of Lloc,T(X), with p: Y ! X ; (iii) a smooth and fiberwise orientation preserving embedding over X , e: D(V ~%) xY S(V -~%) -! M r @M . 58 (It is understood that V in (ii) comes with an involution %; as in (5.12), thi* *s determines ~%: V ! V .) Boundary condition: as in 5.1.3. There is a forgetful map W^T ! WT obtained by passing from V in (ii) of defini* *tion 5.6.6 to h*V , and making the corresponding changes in (iii). By the observations lea* *ding up to definition 5.6.6, the map LT ! WT in definition 5.6.4 has a factorization forget LT _____//_^WT____//WT . (5.* *13) Lemma 5.6.7 The map LT ! W^T in (5.13) is a weak equivalence. Proof An map which is inverse to LT ! W^T up to canonical concordances can be* * defined as follows. Given a surface bundle q :M ! X , an element (p, g, V, h, ffi) 2 Ll* *oc,T(X) and an embedding e as in definition 5.6.6, let Erg= M x R and define Crg+, Crg0, Cr* *g-, C+ , C0 and C- exactly as in definition 5.6.1. Make an embedding Crg+[ Crg0[ Crg--! Erg by (v, t) 7! (e(v), t + g(!(v)), where ! :V ! Y is the vector bundle projectio* *n. Remove from M x R the interior of the image of this embedding, and glue in C+ [ C0 [ C* *- . Call the result E . This comes with a submersion ß :E ! X and a smooth f :E ! R whic* *h is fiberwise Morse. There is also a canonical embedding ~: saddle(V, %) ! E ; to s* *ee this more clearly, reason as in remark 5.6.5. The result is therefore an element of LT(X)* *, consisting of (ß, f), the element (p, g, V, h, ffi) 2 Lloc,T(X) and the embedding ~. * * __|_| Lemma 5.6.8 The forgetful map W^T ! WT in (5.13) is a weak equivalence. Proof The map has a section WT ! W^T. This identifies each set WT(X) with the* * subset of W^T(X) obtained by adding the conditions h = idTxX and g 0 in definition 5* *.6.6(ii). It suffices to show that the section WT ! W^T satisfies the relative surjectivi* *ty criterion of proposition 2.2.6. Let a surface bundle q :M ! X , an element (p, g, V, h, ffi) 2 Lloc,T(X) and an* * embedding e as in definition 5.6.6 be given. The proof of lemma 5.5.2 gives us an explici* *t concordance (~p, ~g, ~V, ~h, ~ffi) 2 Lloc,T(Xx ]0, 1[ ) (5.* *14) from (p, g, V, h, ffi) 2 Lloc,T(X) to an element in the image of Wloc,T(X) ! Ll* *oc,T(X). Here ~p is obtained from p x id:Y x ]0, 1[ -! Xx ]0, 1[ by restriction to an open s* *ubset, and similarly the vector bundle V~ is obtained from V x ]0, 1[ by restriction. It* * is therefore clear that (5.14) lifts to a concordance between elements of W^T(X), in such a * *way that the underlying surface bundle of the concordance is ~q= q x id]0,1[:Mx ]0, 1[ -! Xx ]0, 1[ and the underlying embedding ~eis obtained from e x id]0,1[by restriction. If * *a closed subset C of X is given, and the restriction of (p, g, V, h, ffi) to some open n* *eighborhood U of C is already in Wloc,T(U), then the concordance so constructed is constant o* *n U . __|_| 59 Corollary 5.6.9 The map LT ! WT in definition 5.6.4 is a weak equivalence. * * __|_| This completes the construction of diagram (5.1) and the verification that all * *the vertical arrows in it are weak equivalences. 5.7 Using the concordance lifting property Lemma 5.7.1 For fixed T in K , the forgetful map WT ! Wloc,Thas the concorda* *nce lifting property. Proof Let X be a smooth manifold. Any riemannian vector bundle on T xX x[0, 1* *] with isometric involution is isomorphic to the pullback of a riemannian vector bundl* *e on T x X (with isometric involution) along the projection T x X x [0, 1] ! T x X . Cons* *equently, any concordance starting at an element z of Wloc,T(X) is trivial up to an isomo* *rphism of vector bundles. A choice of such a trivializing isomorphism determines, for eac* *h y 2 WT(X) which lifts z , a lifted concordance starting at y . * * __|_| Now fix an element (V, %) in Wloc,T(?). That is, V is an oriented 3-dimensional* * riemannian vector bundle on T , with a fiberwise isometric involution %. For each t 2 T , * *the dimension of the eigenspace Vt-% is equal to the label of t in 3_. The following is true * *by definition. Lemma 5.7.2 The fiber of the forgetful map WT ! Wloc,Tover (V, %) 2 Wloc,T(?* *) is the sheaf which takes an X in X to the set of all pairs (q, e) where (i) q denotes a smooth graphic bundle M ! X of compact oriented surfaces, subj* *ect to a boundary condition as in definition 5.1.3; (ii) e: D(V %) xT S(V -%) x X -! M r @M is a smooth embedding over X which is fiberwise orientation preserving. * * __|_| Corollary 5.7.3 The fiber of the forgetful map WT ! Wloc,Tover (V, %) 2 Wloc,T* *(?) is weakly equivalent to the sheaf which takes an X in X to the set of all smooth * *graphic bundles q :M ! X of oriented compact surfaces, where each fiber has its (orient* *ed) bound- ary identified with @(S1 x [0, 1] ) q -S(V %) xT S(V -%). Proof To get from data (q, e) as in lemma 5.7.2 to the kind of bundle describ* *ed in corol- lary 5.7.3, delete the interior of im(e) from the total space of the surface bu* *ndle q . To get from a surface bundle M ! X as in corollary 5.7.3 to the data described in lemm* *a 5.7.2, form the union of M and (D(V %) xT S(V -%)) x X along (S(V %) xT S(V -%)) x X .* * __|_| Remark 5.7.4 The description of the (homotopy) fiber in corollary 5.7.3 uses* * only the part of T lying over {1, 2} 3_, since spheres of dimension -1 are empty. 60 6 The connectivity problem 6.1 Overview and definitions The previous section gave us decompositions of W and Wlocinto pieces WS and Wlo* *c,S, respectively, and a description of the homotopy fibers of the forgetful maps WS -! Wloc,S as certain surface bundle theories, cf. corollary 5.7.3. For a given S in K ,* * the surfaces involved are typically not connected, so that the representing space of the fib* *er theory is not directly related to the moduli space whose group completion we are studying* *. In this section we remedy this by showing that upon taking the homotopy colimit over S * *, we can in fact assume that the relevant surfaces are connected. Definition 6.1.1 For X in X let Wc,S(X) WS(X) consist of the triples (q, V,* * e) as in definition 5.1.3, with q :M ! X etc., such that the surface bundle M r im(e) -!* * X has connected fibers. Then Wc,Sis a subsheaf of WS and |Wc,S| is a union of connected components of |* *WS|. The forgetful map from Wc,S to Wloc,Sstill has the concordance lifting property* *. By analogy with corollary 5.7.3, we have the following analysis of its fibers. Corollary 6.1.2 The fiber of the forgetful map Wc,S! Wloc,Sover V 2 Wloc,S(?)* * is weakly equivalent to the sheaf which takes an X in X to the set of all smooth * *graphic bundles q :M ! X of oriented compact connected surfaces, where the boundary of * *each fiber Mx is identified with @(S1 x [0, 1] ) q -(S(V %) xS S(V -%)). It would therefore be nice to have a statement saying that the inclusion of hoc* *olimS Wc,S in hocolimSWS is a weak equivalence. Unfortunately such a statement is nonsensi* *cal if we insist on letting S run through the entire category K . We have a contravariant* * functor S 7! WS from K to the category of sheaves on X , but we do not have a subfunct* *or S 7! Wc,S. It is not the case that the map (k, ")*: WT ! WS induced by a morphism (k, "): S ! T in K will always map the subsheaf Wc,T to* * the subsheaf Wc,S. Let us take a more careful look at this phenomenon. We may assume that k is an inclusion and that T r S has exactly one element t, * *with label ~(t) 2 3_and sign "(t) 2 { 1}. Fix (q, V, e) in WT(X), with q :M ! X and* * let (q0, V 0, e0) be the image of (q, V, e) in WS(X), with q0: M0 ! X . For each x * *2 X there is a canonical embedding of surfaces Mx r im(ex) -! M0xr im(e0x). 61 The complement of its image is identified with D(V(%t,x)) x S(V(-%t,x))if"(t) = +1,and S(V(%t,x)) x D(V(-%t,x))if"(t) = -1, where V(t,x)is the fiber of V over (t, x) 2 T xX . We have a problem when the c* *omplement is nonempty but has empty boundary, because then it will contribute an additional * *connected component. This happens precisely when (~(t), "(t)) = (3, +1) and when (~(t), * *"(t)) = (0, -1). In all other cases, there is no problem. Now our indexing category K is equivalent to a product K03x K12. The categorie* *s K03 and K12 can be described as full subcategories of K : namely, K03 is spanned b* *y the objects S whose reference map S ! 3_has image contained in {0, 3} and K12 is sp* *anned by the objects S whose reference map S ! 3_has image contained in {1, 2}. For homotopy colimits of functors from a product category to spaces (or to shea* *ves on X ) there is a Fubini principle. In our case it states that hocolim WT ' hocolim hocolim WQqS . (6* *.1) T inK Q inK03 S inK12 Lemma 6.1.3 For any morphism (k, "): P ! Q in K03, the commutative square (k,")* hocolim WQqS ________//hocolim WPqS S inK12 S inK12 | | | | | | fflffl| (k,")* fflffl| hocolim Wloc,QqS______//hocolimWloc,PqS S inK12 S inK12 is homotopy cartesian (after passage to representing spaces). Theorem 6.1.4 The inclusion hocolim Wc, S-! hocolim WS S inK12 S inK12 is a weak equivalence. Theorem 6.1.4 is the main result of the section. We develop a surgery method t* *o prove it. The idea is to make nonconnected surfaces connected by means of multiple s* *urgeries on embedded (thickened) 0-spheres. Then we need to know that such multiple 0-su* *rgeries on a surface are essentially unique. In order to state the uniqueness, we view * *them as the objects of a category. 62 6.2 Categories of multiple surgeries Definition 6.2.1 Let M be a compact, smooth, nonempty surface. Let CM be the* * to- pological category defined as follows. An object consists of a finite set T a* *nd a smooth orientation preserving embedding eT of D2 x S0 x T in M r @M , subject to the c* *ondition that surgery on eT results in a connected surface. A morphism from (S, eS) to (* *T, eT) is an injective map k :S ! T such that k*eT = eS . The set of objects ob(CM ) is topologized as a disjoint union, over all T , of * *spaces of smooth embeddings from D2 x S0 x T to M r @M , with the compact-open C1 topology. T* *he total morphism set mor (CM ) is topologized as a closed subset of ob(CM ) x ob(* *CM ) via the map (source,target). Proposition 6.2.2 The space BCM is contractible. The proof requires a lemma. Lemma 6.2.3 Let oe :N ! X be a submersion of smooth manifolds without bounda* *ry, dim (N) > dim(X). Suppose that for each x 2 X there exists a contractible open * *neighbor- hood V of x in X , a finite set Q and a map QxV ! N over X inducing a surjectio* *n from Q ~=ß0(Q x V ) to ß0(Ny) for every y 2 V . Then there exists a locally finite c* *overing of X by contractible open sets Vj, where j 2 J , and finite sets Qj, and a smooth * *embedding a a: Qj x Vj -! N j over X , such that the restriction of a to Qj x Vj induces surjections Qj ! ß0(* *Nx), for each j 2 J and x 2 Vj. Example 6.2.4 The submersion R2 r (0, 0) -! R ; (x, y) 7! x satisfies the hy* *pothesis of lemma 6.2.3. The submersion R r 0 ! R ; x 7! x does not. Surjectivity is not* * directly related to the issue; the projection from (R x {0, 1}) r (0, 0) to R is a surje* *ctive submersion which also fails to satisfy the hypothesis of lemma 6.2.3. Proof of lemma 6.2.3. Note first that the statement is not completely trivial* *. Using the hypothesis, we could start with a locally finite covering of X by contractible* * open sets Vj, and choose finite sets Qj and maps aj: Qj x Vj ! N over X inducing surject* *ions Qj ! ß0(Ny) for every y 2 Vj. This would give us a map a a: Qj x Vj -! N j which is an immersion. Unfortunately there is no guarantee that it is an embedd* *ing. To solve this problem we will partition a äl rge", dense open subset U of N into* * "levels" indexed`by the real numbers, and arrange that a maps distinct connected compone* *nts of Qj x Vj to distinct levels of U . Then a is an embedding. 63 The jet transversality theorem, applied to sections of the vertical tangent bun* *dle of N , implies that we can find a k 0 and a smooth f :N ! R such that the fiberwise * *k-jet prolongation jkfff :N ! Jkff(N, R) is nowhere 0. Let U N consist of all z 2* * N such that f|Nff(z)is regular at z . Then U is open in N and Ux := U \ Nx is dense i* *n Nx, for each x 2 X . Hence the inclusions Ux ! Nx induce surjections ß0(Ux) ! ß0(N* *x). The hypotheses on oe now give us a a covering of X by contractible open subsets* * Vj, and for each Vj a finite set Qj and a map aj: Qj x Vj ! U over X such that the indu* *ced composite map Qj ! ß0(Ux) ! ß0(Nx) is onto for every x 2 Vj. We can assume that* * the Vj are the open stars of the vertices in a sufficiently fine triangulation of X* * , in which case the covering is locally finite. But in addition we can`easily arrange that faj * *is constant on q x Vj for each q 2 Qj, and that the resulting map jQj ! R is injective. Then* * the map a which equals aj on Qj x Vj satisfies all our requirements. * * __|_| In the proof of theorem 6.1.4, we will use a sheaf version CM of CM . For conn* *ected X in X let CM (X) be the (discrete) category whose objects are the pairs (T, eT) wh* *ere T is a finite set and eT :D2 x S0 x T x X -! (M r @M) x X is a smooth embedding over X , fiberwise orientation preserving and subject to * *the condition that fiberwise surgery on eT results in a bundle of connected surfaces. A morph* *ism from (S, eS) to (T, eT) is an injective map k :S ! T such that k*eT = eS . Since ob(CM ( ke)) is the set of smooth maps from keto the embedding space ob(* *CM ), one gets a functor of topological categories |CM | ! CM which induces a degreewise* * homotopy equivalence of the nerves and therefore a homotopy equivalence B|CopM| ~=B|CM |* * ! BCM . (Here it is best to define BCM as the fat realization [35] of the nerve of CM * *, ignoring the degeneracy operators.) Proof of proposition 6.2.2. We show that fiCopMis weakly equivalent to the ter* *minal sheaf taking every X in X to a singleton. By proposition 2.2.6, this reduces* * to the following Claim. Let X in X be given with a closed subset A and a germ s 2 colimUfi* *CopM(U), where U ranges over the neighborhoods of A in X . Then s extends to an el* *ement of fiCopM(X). To verify this, choose an open neighborhood U of A in X such that the germ s ca* *n be represented by some s0 2 fiCopM(U). The information contained in s0 includes a* * locally finite covering of U by open subsets Uj for j 2 J . (Making U smaller if necess* *ary, we can assume that this is locally finite in the strong sense that every x 2 X has a n* *eighborhood which meets only finitely many Uj.) It also includes a choice of object _RR 2 o* *b(CM (UR )) for each finite nonempty subset R of J . (There are also morphisms _RS 2 mor(CM* * (US)), but they are of course determined by their sources _RR |US and targets _SS .) N* *ext, choose an open X0 X such that U [ X0 = X and the closure of X0 in X avoids A. 64 Let N be the open subset of (M r @M) x X0 obtained by removing from (M r @M) x * *X0 the closures of the embedded 2-disk bundles determined by the various 'RR |UR \* * X0. By making U and X0 and the Uj smaller if necessary, but taking care that the Uj re* *main the same near A, we can arrange that the projection N ! X0 satisfies the hypoth* *esis of lemma 6.2.3. By the lemma, there exists a locally finite`covering of X0 by contractible open* * sets Vj, and finite sets Qj and an embedding a of j Qj x Vj in N , over X0 , such that* * a induces surjections Qj ! ß0(Nx) for each j and x 2 Vj. (Again, making X0 small* *er if necessary, we can assume that this is locally finite in the strong sense that e* *very x 2 X has a neighborhood`which meets only finitely many Vj.) We can also choose a sm* *ooth embedding b of jQjx Vj in N , over X0, inducing constant maps Qj ! ß0(Nx) for* * eachfi j and x 2 Vj, and such that im(a)\im (b) = ;. (Forfexample,ithe distinct sheets* * of bfiQjxVj can be chosen very close to a selected sheet of afiQjxVj.) Since the Vj are con* *tractible, the normal bundles of a and b can be trivialized (as oriented 2-dimensional vector * *bundles), and so the ü nionö f a and b extends to a smooth and fiberwise orientation pre* *serving embedding ` c: D2 x S0 x j(Qj x Vj) -! N over X0. For each j with nonempty Vj, the restriction of c to D2 x S0 x Qj x Vj* * is an object 'jj of CM (Vj). Finally we can arrange that Vj is empty whenever Uj is n* *onempty. We are now ready to define an explicit element in fiCopM(X) which extends the g* *erm s. Let Yj = Uj if Uj is nonempty, Yj = Vj if Vj is nonempty, and Yj = ; for all other * *j 2 J . Then the Yj form a locally finite open covering of X . For finite R J with n* *onempty YR , we can write YR = US \ VT for disjoint subsets S, T of R with S [ T = R.* * Let 'RR 2 ob(CM (YR )) be the coproduct (which exists by construction) of _SS|YR * *and the 'jj|YR for j 2 T . The covering j 7! Yj together with the data 'RR for finite * *nonempty R J is an element in fiCopM(X) which extends the germ s. * * __|_| 6.3 Parametrized multiple surgeries We reformulate proposition 6.2.2 in a parametrized setting and deduce theorem 6* *.1.4 from the reformulation. First we remind the reader of Segal's edgewise subdivision o* *f a category. Remark 6.3.1 For any category D , the edgewise subdivision es(D) of D is an* *other category defined as follows. An object of es(D) is a morphism f :c0 ! c1 in D * *. A morphism in es(D) from an object f :c0 ! d0 to an object g :d0 ! d1 is a commut* *ative square c0O__f_//_c1O | | | | | g fflffl| d0 ____//_d1 65 in D . It is well known that B(es(D)) is homeomorphic to BD , if D is a discret* *e category. More precisely, by [13, Lm.2.4] the nerve of es(D) is isomorphic as a simplicia* *l set to the edgewise subdivision of the nerve of D , and this implies by [36] that the * *realizations are homeomorphic. In the case of a simplicial category D one can argue degreewi* *se. The general case of a topological category can in most cases be reduced to the case* * of a simplicial category. Definition 6.3.2 Fix an object S in K12. Let (T, U) be a pair of finite sets w* *ith U T and T \S = ;. We introduce a sheaf WS;(T,U)on X with a forgetful map WS;(T,U)!* * WS . For X in X , an element in WS;(T,U)(X) is an element (q, V, e) of WS(X) with q * *:M ! X etc., together with a smooth embedding eT :D2 x S0 x T x X -! M r @M over X , avoiding im(e). Condition: Fiberwise surgery on eU results in a bundl* *e of con- nected surfaces; here eU denotes the restriction of eT to D2 x S0 x U x X . Let P be the category whose objects are pairs of finite sets (T, U) with U T * *, where a morphism from (Q, R) to (T, U) is an injective map h: Q ! T with h(R) U . Suc* *h a morphism (Q, R) ! (T, U) induces a map of sheaves WS;(T,U)-! WS;(Q,R), so that * *there is a contravariant functor from P to sheaves on X given by (T, U) 7! WS;(T,U). Corollary 6.3.3 The forgetful maps WS;(T,U)! WS induce a homotopy equivalence hocolim |WS;(T,U)| ' |WS| . (T,U) Proof Fix an element in WS(?), consisting of an oriented surface M and a smoo* *th orien- tation preserving embedding e: D(V %) xS S(V -%) -! M r @M. where V denotes a 3-dimensional oriented riemannian vector bundle over S with i* *nvolution. It is enough to show that the homotopy fiber of hocolim |WS;(T,U)| -! |WS| (T,U) over the point corresponding to M is contractible. In this situation the proces* *ses of forming homotopy colimits and homotopy fibers commute. Moreover each of the forgetful * *maps WS;(T,U)! WS has the concordance lifting property, so by proposition A.2.6, the* * homotopy fiber which we are interested in is weakly equivalent to hocolim | fiberM(WS;(T,U)! WS)|. (6* *.2) (T,U) 66 Let MS be the compact surface obtained from M by deleting int(im (e)). It is cl* *ear that each expression | fiberM(WS;(T,U)! WS)| in (6.2) can be replaced by the natural* *ly homo- topy equivalent mor(T,U)CMS , the space of morphisms in CMS of definition 6.2.1 which induce the inclusion U* * ! T of finite sets. The homotopy colimit now becomes the classifying space of the * *transport category Popsmor oCMS , cf. section D.1, where the bullet stands for objects (T, U) of P . This is a ca* *tegory whose objects are the morphisms (U, eU ) ! (T, eT) in CMS where the underlying map U* * ! T is an inclusion. The morphisms correspond to certain commutative squares in CMS . * *What we have here is a category equivalent to the edgewise subdivision (see remark 6.3.* *1 above) of CMS . Its classifying space is therefore homotopy equivalent to BCMS , hence co* *ntractible by proposition 6.2.2. * * __|_| Proof of theorem 6.1.4. Using the homotopy invariance property of homotopy di* *rect limits, we obtain from corollary 6.3.3 a homotopy equivalence of spaces j+ : hocolim hocolim |WS;(T,U)|______//hocolim |WS| . S inK12 (T,U) inP S inK12 We compare this with the map j- : hocolim hocolim |WS;(T,U)|_______//hocolim |Wc,R| S inK12 (T,U) inP R inK12 (6* *.3) induced by the composite maps * WS;(T,U) ______// WS[T ___(-)//_WS[(TrU) (6* *.4) and renaming, S [ (T r U) _ R. Here the first arrow in (6.4) is self-explanato* *ry. The second is induced by the inclusion S [ (T r U) ! S [ T , with the sign function* * on U which is -1. Thus the first arrow amounts to adding the surgery data corresp* *onding to labels in T (but not performing any surgeries), while the second amounts to * *performing the surgeries corresponding to labels in U T . It follows that the composite * *map in (6.4) lands in the subsheaf Wc,S[(TrU), as required in (6.3). The map j- in (6.3) is* * clearly a retraction, with a canonical section i which identifies each Wc,Rwith WR;(;,;).* * The target of j- is contained in the target of j+ , so we may ask whether j- and j+ are ho* *motopic as maps to hocolimS|WS|. This is indeed the case, by remark D.1.3 and the fact * *that each 67 WS;(T,U)fits into a natural commutative diagram WS;(T,U) w | LLLL forgetwwwww| LL(6.4)L LL www || LLLL --www(+)* fflffl|(-)* L%% WS oo______WS[T _______//_WS[(TrU) . The homotopy restricts to a constant homotopy from j+ i to j- i . Consequently* *, it is a deformation retraction of hocolimS |WS| to hocolimS |Wc,S|. * * __|_| 6.4 Annihiliation of 2-spheres The goal is to prove lemma 6.1.3. Most of the proof is based on some elementary* * product decompositions. Lemma 6.4.1 Let T = T1 [ T2 be a disjoint union, where T1 is an object of K0* *3 and T2 is an object of K . There are weak equivalences, natural in T2 for fixed T1 , WT -! Wloc,T1x WT2, Wloc,T -! Wloc,T1x Wloc,T2. Proof The second map is induced by the inclusions T1 ! T and T2 ! T . It shou* *ld be clear that it is a weak equivalence. Note that sign functions on T2 and T1 are * *not needed. The first coordinate of the first map is again induced by the inclusion T1 ! T * *. The second coordinate of the first map, WT -! WT2 , is defined as follows. Let (q, V, e) be an element of WT(X) as in definition 5* *.1.3, with q :M ! X . For a 2 T1 , the bundle D(Va%) xXa S(Va-%) (where Xa = a x X and Va = V |Xa) is either empty or a bundle of 2-spheres. In * *any case it has empty boundary and its image under e is a union of connected components * *of M . Let M0 be obtained from M by deleting these components, for all a 2 T1. Let V 0* *be the restriction of V to T2 x X and let e0 be the restriction of e to a % -% D(Vb ) xXb S(Vb ) . b2T2 Then (q0, V 0, e0) 2 WT2(X). This determines the map WT -! WT2. Again it shou* *ld be clear that the resulting map WT2 -! Wloc,T1x WT2 is a weak equivalence: it is easy to write down an inverse for the induced map * *on homotopy groups. * * __|_| 68 Proof of lemma 6.1.3. We may assume that the complement of P in Q has exactly* * one element. Using lemma 6.4.1, we can isolate a common factor Wloc,Pin each of th* *e four terms of the square, then remove it. In other words, we can also assume that P * *= ;, which implies that Q has exactly one element. (We still have a morphism in K03 from ;* * to Q and we are not going to throw it away.) By the same reasoning we can now isolate a * *common factor hocolim Wloc,S S inK12 in the two terms of the lower row of the square, then remove it. Next we can i* *solate a common factor Wloc,Qin the two terms of the left-hand column. We can again remo* *ve it because Wloc,Qis connected: Wloc,Q[?] is a singleton. This leaves us with a squ* *are of the form hocolim WS - -u-! hocolim WS S inK12 S inK12 ?? ? y ?y ? -------! ? where the map u can be described as follows. Choose a base point z 2 Wloc,Q(?).* * Then Wloc,Qbecomes a sheaf on X with values in the category of pointed sets and giv* *es us a canonical inclusion of WS in Wloc,Qx WS ' WQqS for each S in K12. Compose that with the map WQqS ! WS induced by our morphism ; ! Q, regard S as a variable and apply hocolimS. We have to show that u is a weak equivalence. In order to do that we make a cas* *e distinction. The reference map Q ! {0, 3} 3_amounts to a choice of an element ` from {0, 3* *} and the morphism ; ! Q amounts to a choice of an element m 2 {-1, +1}. Case 1 is the case where (`, m) = (0, +1) or (`, m) = (3, -1). By inspection, * *u is the identity map in that case. Case 2 is the case where (`, m) = (3, +1) or (`, m) = (0, -1). Here we note tha* *t our choice of z 2 Wloc,Q(?) determines an oriented 3-dimensional vector space V with inne* *r product. We can assume V = R3. The map u is given by disjoint union of all surfaces in s* *ight with S2. More precisely, for each S in K12 and X in X , we have a map uS,X: WS(X) ! WS(X) given by (q, V, e) 7! (q], V, e) where q :M ! X is a surface bundle etc., and q* *] is obtained from q by disjoint union with a trivial sphere bundle S2 x X ! X . This is natu* *ral in the variables X and S and so induces u above. Lemma 6.4.2 The map u: hocolim WS - ! hocolim WS S inK12 S inK12 given by disjoint union of all surfaces in sight with S2 is a weak equivalence. 69 Proof Using concatenation of surfaces, we can put a monoid structure on hocol* *imS WS . More precisely, we have for each S and T in K12 a map concatenation:WS x WT ! WSqT and this induces a multiplication hocolim WS x hocolimWT -! hocolim WU . S inK12 T inK12 U inK12 Now let y and y0 be the elements of W;(?) determined by the surfaces (S1 x [0, * *1] ) q S2 and S1x[0, 1], respectively. The map u under investigation is simply given by c* *oncatenation with y , where we use the inclusion W; hocolimWS . S inK12 But the homotopy class of u (after passage to representing spaces) depends only* * on the component of y in fifi fi hocolim WSfi. S inK12 The surgery methods of the previous subsection show immediately that this agree* *s with the component of y0. Hence u is homotopic, after passage to representing spaces, to* * the map given by concatenation with y0. * * __|_| 7 Stabilization 7.1 Stabilization Choose z 2 Vc(?) of genus 1. For every X in X , the unique map X ! ? induces Vc(?) ! Vc(X) and so allows us to think of z as an element of Vc(X). Let z-1Vc * *be the sheaf on X obtained by sheafifying the contravariant functor (alias presheaf) i z. z. z. z. j X -! colim Vc(X) -! Vc(X) -! Vc(X) -! Vc(X) -! . . ., where z. denotes concatenation with z . The sheafification process is very mild* * in this case. In particular, the presheaf and its sheafification agree on compact objects of * *X , such as spheres. Hence the canonical map from i z. z. z. z. j z-1|Vc| = colim |Vc| -! |Vc| -! |Vc| -! |Vc| -! . . . to |z-1Vc| is a homotopy equivalence. Similarly, for each S in K , define z-1WS and z-1Wc,Sas the colimits, in the ca* *tegory of sheaves on X , of the diagrams WS -z.!WS -z.!WS -z.!WS -z.!. .,. Wc,S-z.!Wc,S-z.!Wc,S-z.!Wc,S-z.!. .,. 70 respectively. Then again we have homotopy equivalences |z-1WS| ' z-1|WS| , |z-1Wc,S| ' z-1|Wc,S| . Moreover, since z-1|Vc|, z-1|WS| and z-1|Wc,S| have been defined as sequential * *colimits of CW-spaces, they can also be regarded as homotopy colimits: for example, i z. z. z. z. j z-1|WS| ' hocolim |WS| -! |WS| -! |WS| -! |WS| -! . ... Proposition 7.1.1 B|Vc| ' Z x B 1+,2. ` Proof We noted in section 1 that |Vc| ' gB g,2. It follows that |z-1Vc| ' Z x B 1,2. On the other hand the bar construction gives us a simplicial space Eo with Ek = |z-1Vc| x |Vc|k and a simplicial map from it to k_ 7! ? x |Vc|k . The Harer stability theorem implies that this simplicial map satisfies the hypo* *theses of corollary C.1.2, so that we have a homology fibration sequence |z-1Vc| -! |Eo| -! B|Vc|. It only remains to show that |Eo| ' ? . To this end observe that |Eo| is homoto* *py equivalent to the realization of a monotone union of simplicial spaces of the form k_7! |V* *c|k+1. Each of these has a contractible realization. * * __|_| For an object T in K12, corollary 6.1.2 implies that the homotopy fiber`of the * *localization map |Wc,T| -! |Wloc,T| over any base point is homotopy equivalent to g B g, 2* *+2|T|. Lemma 7.1.2 For T in K12, any homotopy fiber of |z-1Wc,T| -! |Wloc,T| is hom* *otopy equivalent to Z x B 1, 2+2|T|. * * __|_| Finally we have the stabilized versions of lemma 6.1.3 and theorem 6.1.4: Corollary 7.1.3 For any morphism (k, "): P ! Q in K03, the commutative square (k,")* -1 hocolim |z-1WQqS | __________//_hocolim|z WPqS | S inK12 S inK12 | | | | fflffl| fflffl| (k,")* hocolim |Wloc,QqS| __________//_hocolim |Wloc,PqS| S inK12 S inK12 is homotopy cartesian. 71 Corollary 7.1.4 The inclusion hocolim |z-1Wc, T| -! hocolim |z-1WT| T inK12 T inK12 is a homotopy equivalence. Corollaries 7.1.3 and 7.1.4 are about a new homotopy colimit decomposition of |* *W|: Lemma 7.1.5 |W| ' | z-1W| ' hocolim |z-1WT| . T inK Proof Since |W| is group complete, the inclusion |W| ! z-1|W| ' |z-1W| is a h* *omo- topy equivalence. The second homotopy equivalence in the chain follows from |z-* *1WT| ' z-1|WT| and __ hocolim z-1|WT| ' z-1 hocolim |WT| . |_| T inK T inK 7.2 Using the Harer-Ivanov stability theorem Lemma 7.2.1 The canonical map from Z x B 1,2 to the homotopy fiber (over the* * base point) of the forgetful map hocolim |z-1Wc,S| -! hocolim |Wloc,S| S inK12 S inK12 induces an isomorphism in homology with integer coefficients. Proof For the object S = ; of K12, we have |z-1Wc,S| ' Z x B 1,2 and |Wloc,S|* * = ?, so that there is indeed a canonical map from Z x B 1,2 to the homotopy fiber of hocolim |z-1Wc,S| -! hocolim |Wloc,S|. S inK12 S inK12 We now check that the hypothesis of corollary C.1.3 is satisfied. Let (k, "): S* * ! T be a morphism in K12. We have to verify that, in the commutative square of spaces |z-1Wc,T| _____//|Wloc,T| (k,")*|| (k,")*|| fflffl| fflffl| |z-1Wc,S| _____//|Wloc,S|, the induced map from any of the homotopy fibers in the upper row to the corresp* *onding homotopy fiber in the lower row induces an isomorphism in homology. The homotop* *y fibers in question are related by a map Z x B 1, 2+2|T|-! Z x B 1, 2+2|S| 72 given geometrically by attaching cylinders D1 x S1 or double disks D2 x S0 to t* *hose pairs of boundary circles which correspond to elements of T r k(S). This map is an i* *ntegral homology equivalence by the Harer-Ivanov stability theorem. Apply corollary C.1* *.3. __|_| Corollary 7.2.2 The canonical map from ZxB 1,2 to the homotopy fiber (over the* * base point) of the forgetful map hocolim |z-1WS| -! hocolim |Wloc,S| S inK S inK induces an isomorphism in homology with integer coefficients. Proof Combine lemma 7.2.1 with corollaries 7.1.4 and 7.1.3. * * __|_| Proof of theorem 1.3.4. By lemma 7.1.5 and diagram 5.1, we have hocolim |z-1WS| ' |W| , hocolim |Wloc,S| ' |Wloc| . S inK S inK Therefore corollary 7.2.2 implies that the homotopy fiber of |W| ! |Wloc| is ho* *mology equivalent to Z x B 1,2. On the other hand, |W| and |Wloc| are infinite loop sp* *aces by theorems 1.3.1 and 1.3.2, and the map |W| ! |Wloc| is an infinite loop map. He* *nce its homotopy fiber is an infinite loop space, hence group complete. It follows that* * the homotopy fiber is Z x B 1+,2' B|Vc|. * * __|_| A More about sheaves A.1 Concordance and the representing space Let F be a sheaf on X . We shall construct a natural transformation #: [X, |F| * *] -! F[X], and an inverse , :F[X] ! [X, |F| ] for #. We start with the construction of , . Fix X in X and an element u 2 F(X). Ch* *oose a smooth triangulation of X , with vertex set T . Suppose that S T is a disti* *nguished subset (the vertex set of a simplex in the triangulation). Let e(S) = {w :S ! R | sw(s) = 1} (S) = {w 2 e(S) | w 0}. The triangulation gives us characteristic embeddings cS : (S) ! X , one for eac* *h distin- guished S T . By induction on S , we can choose smooth embeddings ce,S: e(S) ! X , extending the cS , which are compatible: i.e., if S is distinguished and R S * *is nonempty, then ce,Sagrees with ce,R on eR) e(S). Let uS = ce,S*(u) 2 F( e(S)). Finally choose a total ordering of T . This leads to an identification of each * * e(S) with a 73 standard extended simplex. Consequently it promotes each uS to a simplex of the* * simplicial set n_7! F( ne). We then have a unique map ,(u): X ! |F| such that, for each S* * as above with |S| = n + 1, the diagram ~= (S) ____//_ n cS|| |char.|map foru fflffl|,(fflffl|u) X ______//|F| commutes. It is straightforward to show that the resulting homotopy class of ma* *ps X ! |F| depends only on the concordance class of u 2 F(X). Next we construct #: [X, |F| ] -! F[X]. We may replace |F| by the äf t" realiz* *ation kFk, which is obtained by forgetting the degeneracy operators in n_7! F( ne) and realizing the resulting incomplete simplicial set. So we start with a choi* *ce of map g :X ! kFk. Without loss of generality, we may assume that g is simplicial for * *a smooth triangulation of X with totally ordered vertex set T . That is, for each n 0* * and each distingushed S T , with characteristic map cS : (S) ! X , the composition cS g n ~= (S) _____//_X____//_kFk is the characteristic map associated with some uS 2 F( ne). Choose a smooth homotopy of smooth maps ht: X ! X , where 0 t 1, such that (1) the map h0 is the identity, (2) for every t, the map ht maps each simplex of the triangulation to itself * *and (3) each simplex of the triangulation has a neighbourhood in X which is mappe* *d to the simplex by h1. Then for each n 0 and each distinguished S with |S| - 1 = n and a sufficientl* *y small neighborhood VS of cS( (S)) in X , we obtain a smooth map VS ! e(S) ~= neby composing h1|VS with the inclusion of (S) in e(S) ~= ne. Using this map to pu* *ll back uS 2 F( ne), we obtain compatible elements u0S2 F(VS) which, by the sheaf prope* *rty of F , determine a unique element #(g) of F(X). Again, it is straightforward to ve* *rify that the concordance class of #(g) depends only on the homotopy class of g . Proposition A.1.1 The maps , and # are reciprocal inverses. Proof Let u 2 F(X). We want to show that #,(u) is concordant to u. With sui* *table choices in the constructions above, we have VS im(ce,S) for all distinguished* * S , and then #,(u) equals h1*(u), where (ht: X ! X)0 t 1 is the homotopy which appears * *in the 74 definition of #. Since h1 is smoothly homotopic to h0 = idX, this implies that * *#,(u) is indeed concordant to u. Therefore #, = id:F[X] -! F[X]. In order to show that ,# is the identity on [X, |F| ], we introduce a simplicia* *l monoid Qo whose realization acts on |F|. Namely, Qn is the monoid of smooth maps f : ne! * * ne taking each (extended) face of neto itself. Then Qn acts on the right of F( ne* *) by s .f = f*(s), and so |Qo| acts on |F|. Consequently the monoid [X, |Qo| ] acts on the right o* *f [X, |F| ]. The effect of ,# on an element [g] 2 [X, |F| ] can be described in terms of thi* *s action. Indeed, ,#[g] = [g] . [w] for some w :X ! |Qo|. The map w is determined by h1: X ! X constructed above as part of a homotopy of maps from X to X . Since we are assuming that h1 maps* * the image of each ce,S to itself, ce,S-1h1ce,S is defined. This gives us for each * *n 0 and each n-simplex in the triangulation of X an element in Qn, hence a map from X t* *o the realization of Qo. Since |Qo| is contractible, [w] 2 [X, |Qo| ] is always the n* *eutral element, so that [g] . [w] = [g]. * * __|_| The discussion above has a compact support version as follows. Fix z 2 F(*), so* * that F becomes a functor from X opto pointed sets. For X in X , we will say that an el* *ement s 2 F(X) has compact support if its image in F(X r K) is the base point, for so* *me compact K X . A concordance between elements s0, s1 of F(X) with compact supp* *ort is said to have compact support if it restricts to a constant concordance betwe* *en elements of F(X r K), for some compact K X . The set of compactly supported elements in F* *(X) modulo compactly supported concordance is denoted Fc[X]. Similarly, a map X ! |* *F| is said to have compact support if its restriction to X r K is constant with value* * z , for some compact K X . We let map c(X, |F| ) be the set of such maps. Then we have an * *obvious extension of the above proof: Proposition A.1.2 There is a bijection Fc[X] ~=ß0map c(X, |F| ). * * __|_| This proves proposition 2.2.5 in the special case where A = {z}. The general ca* *se is very similar. * * __|_| A.2 Categorical properties Proposition A.2.1 The construction F 7! |F| takes pullback squares of sheaves* * to pull- back squares of compactly generated Hausdorff spaces. In particular it respects* * products. 75 Proof This is obvious from the definition of |F|. * * __|_| Definition A.2.2 The categoricalQcoproduct F1qF2 of two sheaves F1 and F2 on * *X can be defined by (F1 q F2)(X) = iF1(Xi) q F2(Xi) where Xi denotes the path compo* *nent of X corresponding to an i 2 ß0(X). Proposition A.2.3 |F1 q F2| ~=|F1| q |F2|. Proof Note that neis path-connected for n 0. * * __|_| Definition A.2.4 A natural transformation u: F ! G of sheaves on X has the c* *oncord- ance lifting property if, for X in X and s 2 F(X), any concordance h 2 G(Xx ]0* *, 1[ ) starting at u(s) lifts to a concordance H 2 F(Xx ]0, 1[ ) starting at s. Example A.2.5 Given a natural transformation u: F ! G of sheaves on X , make* * a new sheaf F] as follows. An element of F](X) is a triple (h, s0, s1) where s0 2 F(X* *), s1 2 G(X) and h is a concordance from u(s0) to s1. Then it is not hard to show that the f* *orgetful transformations F] ! F and F] ! G given by (h, s0, s1) 7! s0 and (h, s0, s1) 7* *! s1, respectively, have the concordance lifting property. It is also clear from prop* *osition 2.2.5 that the forgetful map F] ! F defined by (h, s0, s1) 7! s0 is a weak homotopy e* *quivalence. Proposition A.2.6 Suppose given sheaves E, F, G on X and morphisms (alias na* *tural transformations) u: E ! G , v :F ! G . Let E xG F be the fiber product (pullbac* *k) of u and v . If u has the concordance lifting property, then the projection E xG F !* * F has the concordance lifting property and the following square is homotopy cartesian: |E xG F|_____//|F| | | | v| fflffl|u fflffl| |E|________//|G|. A.3 Relative homotopy and fibrations We begin with a special case of proposition A.2.6. Given a natural transformati* *on u: E ! G of sheaves on X with the concordance lifting property, let z be a point in G(?* *) and let Ez be the fiber of u over z (in the category of sheaves). Let hofiberz|u| denote t* *he homotopy fiber of |u|: |E| ! |G| over the point z . Lemma A.3.1 For any y 2 Ez(?), the homotopy set ßn(Ez, y) is in canonical b* *ijection with ßn(hofiberz|u|, y) . 76 Proof (sketch): Because of the concordance lifting property, ßn(Ez, y) can be* * identified with an appropriate relative homotopy group (or homotopy set) of the map of she* *aves u: E ! G . Representatives of the latter are elements (r, s) 2 G(Bn+1) x F(Sn), where Bn+1 = Dn+1 r Sn , such that s 2 F(Sn) is based at y and (r, u(s)) belong* *s to G(Dn+1) G(Bn+1) x G(Sn). See definition 2.2.1. We can identify this relative * *homotopy group (set) with a relative homotopy group (set) of the map of spaces |u|: |E| * *! |G|, which can then be identified with a homotopy group (set) of the homotopy fiber of |u|* * over z . __|_| Corollary A.3.2 In the situation of lemma A.3.1, the sequence " |u| |Ez|Ø___//_|E|___//|G| is a homotopy fiber sequence. Proof The composite map from |Ez| to |G| is constant. This leads to a canoni* *cal map from |Ez| to the homotopy fiber of |u|: |cE| ! |G| over z . It is easy to verif* *y directly that this induces a surjection on ß0. For each y 2 Ez(?), the induced map of homotop* *y sets ßn(Ez, y) -! ßn(hofiberz|u|, y) is the one from lemma A.3.1. It is therefore always a bijection. * * __|_| Proof of proposition A.2.6. We fix z 2 F(?) and obtain v(z) 2 G(?). The fiber* * of E xG F -! F over z is identified with the fiber of u: E ! G over v(z). Using corollary A.3* *.2 we can conclude that the homotopy fiber of |E xG F| -! |F| over z maps to the homotopy* * fiber of |u|: |E| ! |G| over v(z) by a homotopy equivalence. * * __|_| B Sheaves with a category structure This section contains the proof of theorem 4.1.3. B.1 Cocycle sheaves without indices We begin with the definition of a close relative fi0F of fiF . In the definiti* *on of fi0F we trade the open coverings in the definition of fiF for surjective 'etale maps. R* *ecall a smooth map is 'etale if it is locally diffeomorphic, i.e., if its differential at any * *point of the source is invertible. A covering of a smooth manifold X by open subsets Yj, for j 2 J , g* *ives rise to such a map Y ! X , where Y J x X is the set of all (j, x) with x 2 Yj. 77 Definition B.1.1 For X in X , an element in fi0F(X) consists of the following * *data: (i) A smooth manifold Y and a graphic, surjective and 'etale map Y ! X . (We w* *ill write Y (n_)for the manifold of all maps n_! Y such that the composition n_! Y* * ! X is constant.) (ii) For m, n 0 and each injective g :m_ ! n_(which need not be order-preserv* *ing), a morphism 'g in the category F(Y (n_)). The morphisms 'g are subject to a 1-cocycle condition, which comes in two parts* *. The first part says that 'g is an identity morphism if g is bijective. The second p* *art says 'gf = g*('f) O 'g. Here f and g can have the form f :`_! m_and g :m_! n_, so that gf is defined. W* *e have written g*: F(Y (m_)) ! F(Y (n_)) for the map induced by g*: Y (n_)-! Y (m_). Suppose that (Y, '?) is an element of fi0F . Suppose also that Y is an open s* *ubset of J x X and the surjective 'etale map Y ! X which is part of the data has been ob* *tained by restricting the projection J x X ! X . Then Y determines an open covering * *of X by open subsets Yj. Namely, Yj can be defined as the image of Y \ (j x X) unde* *r the projection J x X ! X . Each Y (n_)can be identified with a disjoint union of co* *pies of open subsetsTYS X , where S is a nonempty subset of J with at most n + 1 elements,* * and YS = j2SYj as usual. In this way, '? breaks up into data 'RS 2 F(YS), one fo* *r each pair of finite nonempty R, S J with R S . We leave the detailed verificati* *on to the reader. The conclusion is that fiF is a subsheaf of fi0F . Proposition B.1.2 The inclusion of fiF in fi0F is a weak homotopy equivalence. For the proof we need two lemmas, mostly about 'etale maps. Lemma B.1.3 Let Y ! X be a smooth, 'etale and surjective map. There exists * *an open subset Y [ J x X such that the projection Y [! X is locally finite and surject* *ive, and a map a: Y [! Y over X . In addition, suppose given a closed C X , an open YC[ J x C such that the pr* *ojection YC[! C is locally finite and surjective, and a map aC :YC[! Y over X . Then w* *e can construct Y [and a: Y [! Y above in such a way that Y [|C = YC[and aC = a|YC[. Proof For the absolute case, select a locally finite open covering of X by op* *en subsets Y [j, where j 2 J , such that Y ! X admits a section sj over each Y [j. Let Y [= {(j, x) 2 J x X | x 2 Y [j}. The sj together define a map Y [! Y over X . In the relative case we make a case distinction. If j 2 J is such that YC[ has* * nonempty intersection with j x X , select an open Y [j X in such a way that Y ! X admi* *ts a 78 section sj over Y [jand Y [j\ C equals the image of (YC[) \ (j x X) in X . For * *all other j 2 J , select an open Y [j X r C in such a way that Y ! X admits a section sj* * over Y [j. This is to be done in such a way that the Y [jconstitute a covering of X * *. Then let Y [= {(j, x) 2 J x X | x 2 Y [j} as before. * * __|_| Lemma B.1.4 Let (Y [, '[?) and (Y, '?) be elements of fi0F(X). Suppose also* * that there exists a map g :Y [! Y over X such that g*'? = '[?. Then (Y, '?) and (Y [, '[* *?) are concordant. Moreover, if C X is closed and g is an identity map over a neighborhood of C * *, then the concordance can be constructed so as to be constant over a neighborhood of C . Proof Absolute case: We ignore minor set-theoretic issues related to definiti* *on 2.1.1. _ We want to make a concordance of the form (Z, _?) 2 fi0F(Xx ]0, 1[ ) where Z is the disjoint union of Y [x ]0, 2=3[ and Y x ]1=3, 1[ . This comes wi* *th an obvious surjective 'etale map to Xx ]0, 1[ . There is also a map q :Z ! Y over X defin* *ed by æ [ q(y, t) := g(y) for (y, t) 2 Y x ]0, 2=3[ , y for(y, t) 2 Y x ]1=3, 1[ . We let _? = q*'?. Relative case: We proceed somewhat differently. We are assuming Y [|U = Y |U wh* *ere U is an open subset of X containing C . Also, g equals id on Y [|U . We make a co* *ncordance of the form (Z, _) 2 fi0F(Xx ]0, 1[ ) where Z is the disjoint union of (Y [x ]0, 1=2[ ) [ (Y |U x ]0, 1[ ) [ (Y x ]1=2, 1[ ) and (Y [|(XrC))x ]1=4, 3=4[ . This comes with an obvious surjective 'etale map * *to Xx ]0, 1[ . There is also a map q :Z ! Y over X given by 8 [ >>>: y for(y, t) 2 Y x ]1=2, 1[ , g(y) for(y, t) 2 (Y [|(X r C))x ]1=4, 3=4[ . Again we let _? = q*'?. * * __|_| Proof of B.1.2. This is now a direct consequence of the relative surjectivity * *criterion in proposition 2.2.6 and the two lemmas just above. * * __|_| 79 B.2 A variation on the nerve construction For n 0 let Dn_ be the set of nonempty subsets of {0, 1, 2, . .,.n}. This is* * partially ordered by inclusion. There are functors vn :Dn_! n_given by vn(S) = max (S) 2 * *n_. Lemma B.2.1 Let C be a small category. Then the map of simplicial sets ( n 7! hom (n_op, C )) -! ( n 7! hom (Dn_op, C )) given by composition with vo induces a homotopy equivalence of the geometric re* *alizations. Proof The simplicial set (n 7! hom (Dn_op, C )) is obtained by applying Kan's* * functor ex, the right adjoint of the barycentric subdivision, to (n 7! hom (n_op, C )). The* * statement is therefore a special case of [22, 3.7]. * * __|_| We note that the simplicial set (n 7! hom (n_op, C )) is precisely the nerve of* * C , denoted NoC in section 4. Corollary B.2.2 Let m 7! Cm be a simplicial category. The map of bisimplicia* *l sets (m, n) Ø____//_hom(n_op, Cm ) || || ff'|| (m, n) Ø___//_hom(Dn_op, Cm ) given by composition with the functors vn :Dn_! n_induces a homotopy equivalenc* *e of the geometric realizations. * * __|_| B.3 Completion of the proof Continuing with the proof of 4.1.3, we come to a user-friendly description of t* *he classifying space B|F|. Recall that ne= {(x0, x1, . .,.xn) 2 Rn+1 | ixi= 1}. Lemma B.3.1 The classifying space B|F| is homotopy equivalent to the geomet* *ric real- ization of the simplicial set given by n 7! hom (Dn_op, F( ne)). Proof We defined B|F| in section 4 as the geometric realization of a simplici* *al space given by n_7! | hom(n_op, F)|, where hom (n_op, F) is viewed as a sheaf on X and the* * vertical bars around it indicate the representing space construction. This means that we hav* *e defined B|F| as the geometric realization of a bisimplicial set (m, n) 7! hom (n_op, F( me)). 80 By corollary B.2.2, we may instead use the geometric realization of the bisimpl* *icial set (m, n) 7! hom (Dn_op, F( me)). It is well known that the geometric realization of a bisimplicial set is homeom* *orphic to the geometric realization of its diagonal. In our situation this is the simplicial * *set given by n 7! hom (Dn_op, F( ne)). __|* *_| Lemma B.3.2 Let S be an infinite set and let Zo be a simplicial set. The ge* *ometric real- ization |Zo| is homotopy equivalent to the geometric realization of the incompl* *ete simplicial set n 7! Zn x emb(n_, S), where emb (n_, S) is the set of injective maps from * *n_to S . Proof There is a projection map from the realization of n 7! Zn x emb(n_, S) * *to |Zo|. We will show that it has contractible fibers. Let y be a point in a k-cell of |Zo|* *, corresponding to some nondegenerate simplex in Zk. The fiber over y is homeomorphic to the cl* *assifying space of the poset P whose elements are the nonempty finite subsets of S equipp* *ed with a total ordering and a surjection to k_. For each finite subset P0 of P , there* * exists T 2 P such that sup(T, T 0) exists for all T 02 P0. This implies that the inclusion * *of |P0| in |P| is homotopic to a constant map (with value equal to the vertex determined b* *y T ). Therefore P is contractible, i.e., the fiber in question is contractible. * * __|_| Corollary B.3.3 Let J be the (fixed) infinite set from definition 4.1.1. The* * classifying space B|F| is homotopy equivalent to the geometric realization of the incomplet* *e simplicial set Zo given by n 7! hom (Dn_op, F( ne)) x emb(n_, J). We come to the construction of a comparison map from the incomplete simplicia* *l set in corollary B.3.3 to the simplicial set n_7! fiF( ne). The idea is simple. An n-* *simplex in the incomplete simplicial set of corollary B.3.3 consists of a functor ': Dn_op-! F( ne) and an injective map ~: n_! J . The functor ' carries exactly the same informat* *ion as an element in fiF( ne) whose underlying J -indexed open covering is given by j * *7! neif j = ~(t) for some t 2 n_and j 7! ; otherwise. To make this information more fun* *ctorial, i.e., compatible with face operators, we replace the nonempty open sets in the * *open covering by smaller ones, according to the rule j = ~(t) 7! { (x0, x1, . .,.xn) 2 ne| xt> 0}. (B* *.1) The remaining data can be restricted and we now have an element (', ~) 2 fiF( * *ne). The construction respects the face operators. We now restate theorem 4.1.3 as Lemma B.3.4 The map induces a homotopy equivalence from B|F| ' |Zo| to |f* *iF|. 81 Proof Let z be a vertex of |Zo|. For each n 0, the map induces a map ßn( |Zo|, z) -! ßn(fiF, z) ~=ßn(fi0F, z). We will show that this is bijective by constructing the inverse map. As in sect* *ion A.1, we can represent elements of ßn(fi0F, z) by elements of fi0F(Rn) with compact supp* *ort. Let (Y, '?) be such an element of fi0F(Rn), with notation as in definition B.1.1. T* *here exists a smooth triangulation of Rn , with vertex set T , which for each v 2 T allows an* * embedding gv: st2(v) ! Y over Rn. Here st2(v) is the union of the open stars st(w) of al* *l vertices w adjacent to v . Choose such a triangulation and such embeddings gv. Also, for e* *ach finite nonempty subset S of T spanning a simplex of the triangulation, choose a smoot* *h map ce,S: e(S) ! Rn extending the characteristic inclusion cS : (S) ! Rn. This is * *to be done in such a way that ce,Sagrees with ce,R on a face e(R) e(S) and ce,S( e(S)) st2(v) whenever v 2 S . We then have, for each S as above, a commutative square S x e(S) _____//_Y | | | | fflffl|ce,S fflffl| e(S) _______//Rn where the top row is given by (v, x) 7! gv(ce,S(x)) and the vertical arrows are* * 'etale and surjective. Using this to pull back the data '?, we obtain for each S an element xS 2 hom (D(S)op, F( e(S))) where D(S) is the poset of nonempty subsets of S . Finally we choose a total or* *dering on T and an injection T ! J . This promotes each xS to an element of hom (Dn_op, F* *( ne)) where n = |S| - 1. For each S we also get a canonical injection uS from n_~=S t* *o T J , so that the pair (xS, uS) can be regarded as an n-simplex of Zo. Now we have a * *unique map from Rn to |Zo| which, on (S) Rn, is the characteristic map for the simp* *lex (xS, uS). It has compact support. Its compactly supported homotopy class depend* *s only on the compactly supported concordance class of (Y, '?). This gives us the map : ßn(fi0F, z) -! ßn( |Zo|, z) ~= ßn(B|F|, z) which we need. The composition * : ßn(fi0F, z) ! ßn(fi0F, z) is the identity. Namely, #-1 * * *= #-1 by construction, where # is the bijective map of propositions 2.2.5 and section A. To show that * is the identity on ßn( |Zo|, z), we resurrect the simplicial m* *onoid Qo which was introduced in section A.1. We may replace |Zo| by the geometric reali* *zation of the (complete) simplicial set n 7! hom (Dn_op, F( ne)) on which |Qo| acts (from the right). In this way we get a right action of the m* *onoid ßn|Qo| on ßn( |Zo|, z), with monoid structure coming from that on |Qo|. For every [g] * *2 ßn( |Zo|, z) 82 we have *[g] = [g] . [h1] for some [h1] 2 ßn|Qo| which may depend on [g]. But |Qo| is contractible, so [* *h1] will always be the neutral element and [g] . [h1] = [g]. * * __|_| C Geometric realizations and the bar construction C.1 Realization, quasifibrations and homology fibrations Lemma C.1.1 Let uo: Eo -! Bo be a map between incomplete simplicial spaces * *(or good simplicial spaces). Suppose that the squares uk Ek ______//_Bk di|| |di| fflffl|uk-fflffl|1 Ek-1 ____//_Bk-1 are all homotopy cartesian (k i 0). Then the following is also homotopy car* *tesian: u0 E0 ______//_B0 |incl.| |incl.| fflffl||uofflffl|| |Eo|______//|Bo|. Proof It suffices to prove the statement for incomplete simplicial spaces. (* *The realiza- tion of a good simplicial space is homotopy equivalent to the realization of th* *e underlying incomplete simplicial space.) Without loss of generality all the maps uk: Ek !* * Bk are fibrations. Then, by inspection, |uo| from |Eo| to |Bo| is a quasifibration in * *the sense of [6]. By [6], [7], this implies that each fiber of |uo| maps by a weak homotopy * *equivalence to the corresponding homotopy fiber. Hence the canonical map from E0 to the hom* *otopy pullback of |uo| incl. |Eo|_____//_|Bo|oo__B0 is a weak homotopy equivalence. * * __|_| Corollary C.1.2 Let uo: Eo -! Bo be a map between incomplete simplicial space* *s (or good simplicial spaces). Suppose that, in each square uk Ek ______//_Bk di|| |di| fflffl|uk-fflffl|1 Ek-1 ____//_Bk-1 83 the canonical map from any homotopy fiber of uk to the corresponding homotopy f* *iber of uk-1 induces an isomorphism in integer homology. Then in the square u0 E0 ______//B0 |incl.| incl.|| fflffl||uofflffl|| |Eo|____//_|Bo|, the canonical map from any homotopy fiber of u0 to the corresponding homotopy f* *iber of |uo| induces an isomorphism in integer homology. Proof It suffices to prove the statement for incomplete simplicial spaces. Ag* *ain we may assume that each uk: Ek ! Bk is a fibration. Let D be the functor X 7! S1 ^ X+ from spaces to pointed spaces. Let D(Ek; uk) be the result of applying D to eac* *h fiber of uk: Ek ! Bk. We still have quasifibrations D(Ek; uk) -! Bk and we are in a situation where the previous lemma can be applied; so we get a * *homotopy cartesian square D(E0; u0)______//B0 incl.|| incl.|| fflffl| fflffl| |D(Eo; uo)|___//_|Bo| where the horizontal arrows are quasifibrations. But the lower left hand term * *is homeo- morphic to D(|Eo|; |uo|), the space obtained by applying D fiberwise to the fib* *ers of |uo|. Hence we may ü ndo" the D operation in the left-hand column, replacing D(E0; u0* *) by E0 and |D(Eo; uo)| ~=D(|Eo|; |uo|) by |Eo|, without changing the homology of the h* *orizontal fibers except for a degree shift. * * __|_| Corollary C.1.3 Let C be a small category and let u: G1 ! G2 be a natural tra* *nsforma- tion between functors from C to spaces. Suppose that, for each morphism f :a ! * *b in C , the map f* from any homotopy fiber of ua to the corresponding homotopy fiber of* * ub in- duces an isomorphism in integer homology. Then for each object a of C , the inc* *lusion of any homotopy fiber of ua in the corresponding homotopy fiber of u*: hocolimG1 ! hoc* *olimG2 induces an isomorphism in integer homology. ` ` Proof Apply corollary C.1.2 with Ek := G1(D(k)) and Bk = G2(D(k)), where * *both coproducts run over the set of contravariant functors D from the poset k_to C .* * Then |Eo| is hocolimG1 and |Bo| is hocolimG2. * * __|_| 84 C.2 The bar construction for monoids without unit Let A be a topological monoid, not necessarily with unit. This determines an in* *complete simplicial space Xo where Xk = Ak and the face operators di: Xk ! Xk-1 are give* *n by (a1, a2, . .,.ak)7!(a1, a2, . .,.ai-1, aiai+1, ai+2, . .,.ak)ifi 6= 0, k (a1, a2, . .,.ak)7!(a2, a3, . .,.ak) ifi = 0 (a1, a2, . .,.ak)7!(a1, a2, . .,.ak-1) ifi = k. Of course, Xo is known as the bar construction on A and |Xo| is known as the cl* *assifying space of A. If A has a unit (neutral element), we can use it to define degeneracy operators* * in Xo, making Xo into a simplicial space. If this is a good simplicial space [35], the* *n its realization as a simplicial space is homotopy equivalent to the realization of the underlyi* *ng incomplete simplicial space. Either of these two realizations can therefore be regarded as* * the classifying space of A. A topological monoid A, with or without unit, determines another topological mo* *noid A+ which, as a space, is the disjoint union of A with a singleton. The added point* * serves as the neutral element (unit) in a topological monoid structure on A+ which extend* *s the one on A. (If A did have a unit to begin with, then that will no longer be the unit* * in A+ , but of course it will be a central idempotent in A+ .) Lemma C.2.1 The simplicial space k 7! (A+ )k is good and its realization (a* *s a complete simplicial space) is homeomorphic to the realization of the incomplete simplici* *al space k 7! Ak. Proof It is well known that the forgetful functor from complete simplicial sp* *aces to incom- plete simplicial spaces has a left adjoint. We denote it by Xo 7! X~o. The most* * common description of X~o is as follows: ak a X~k:= Xm m=0 f :k_``m_ where f runs through all surjective order preserving maps from k_to m_. This ma* *kes it fairly clear how X~o is a simplicial space. Namely, suppose given an order preserving * *g :j_! k_ and a triple (m, f, x) in X~k, so that f :k_! m_is order preserving and x 2 Xm * *. Then we let g*(m, f, x) := (m0, v, u*x) where fg = uv is the unique decomposition of fg :j_! m_into an order preserving* * surjec- tion v :j_! m_0and an order preserving injection u: m_0! m_. The simplicial space X~ois good and its geometric realization as a complete sim* *plicial space is homeomorphic to the geometric realization of the incomplete simplicial space* * Xo. 85 If Xo is the bar construction on A, that is, Xk = Ak, then X~o becomes the bar * *construc- tion on A+ , that is, X~k= (A+ )k. Therefore k 7! (A+ )k is a good simplicial * *space and its realization as such is homeomorphic to the realization of the incomplete si* *mplicial space k 7! Ak. * * __|_| The observations above concerning topological monoids with or without unit can * *be general- ized to topological categories with or without identity morphisms. (Monoids are* * categories with only one object.) The category version of lemma C.2.1 is implicit in [25, * *x2.1]. D Generalities about homotopy colimits and stratifications D.1 Homotopy colimits Any functor D from a small (discrete) category C to the category of spaces has * *a colimit, colim D. This is the quotient space of the coproduct a D(a) a inC obtained by identifying x 2 D(a) with f*(x) 2 D(b) for any morphisms f :a ! b i* *n C and elements x 2 D(a). It is well known that the colimit construction is not we* *ll behaved from a homotopy theoretic point of view. Namely, suppose that w :D1 ! D2 is a n* *atural transformation between functors from C to spaces and that wa : D1(a) ! D2(a) i* *s a homotopy equivalence for any object a in C . Then this does not in general impl* *y that the map induced by w from colimD1 to colimD2 is again a homotopy equivalence. Example D.1.1 Let C be the poset of proper subsets of {0, 1}, ordered by in* *clusion. The diagrams [0, 1] - @[0, 1] ,! [0, 1], ? @[0, 1] ,! ? can be regarded as functors from C to spaces. There is a natural transformatio* *n w from the first to the second such that wa is a homotopy equivalence for each object * *a in C . The colimit of the first diagram is homeomorphic to S1. The colimit of the second d* *iagram is a single point. Call a functor D from C to spaces cofibrant if, for any diagram of functors (f* *rom C to spaces) and natural transformations D __v__//Eowo_F where wa: F(a) ! E(a) is a homotopy equivalence for all a 2 C , there exists a * *natural transformation v0: D ! F and a natural homotopy D(a) x [0, 1] ! E(a) (for all a* *) con- necting wv0 and v . It is not hard to show the following. If v :D1 ! D2 is a * *natural transformation between cofibrant functors such that va: D1(a) ! D2(a) is a homo* *topy 86 equivalence for each a 2 C , then v has a natural homotopy inverse (with natura* *l homotop- ies) and therefore the induced map colimD1 ! colimD2 is a homotopy equivalence. This suggests the following procedure for making colimits homotopy invariant. S* *uppose that D from C to spaces is any functor. Try to find a natural transformation D0! D s* *pecializing to homotopy equivalences D0(a) ! D(a) for all a in C , where D0 is cofibrant. T* *hen define the homotopy colimit of D to be colimD0. If it can be done, hocolim D is at le* *ast well defined up to homotopy equivalence. (If D is the second diagram in example D.1.* *1, then the first diagram in the same example can serve as D0 because it happens to be * *cofibrant. This gives hocolimD ~=S1.) This point of view is carefully presented in [8]. Some of the ideas go back to * *[26]. As we will see in a moment, there is a canonical construction for D0 which depends na* *turally on D. The standard foundational reference for homotopy colimits and homotopy limits i* *s the book [3] by Bousfield and Kan. But the first explicit construction of homotopy colim* *its in general appears to be due to Segal [37]. Again let D be a functor from a discrete small category C to the category of s* *paces. Following Segal we introduce a topological category denoted C sD, the transport* * category of D: a a ob(C sD) = D(a) , mor(C sD) = D(oe(f)) . a2ob(C ) f2mor(C ) Here oe(f) denotes the source of a morphism f in C . We will write morphisms in* * C sD as pairs (f, x) where f 2 mor (C ) and x 2 D(oe(f)). The composition (g, y) O * *(f, x) of two such morphisms is defined if an only if g O f is defined in C and f*(x) = y* * , in which case (g, y) O (f, x) = (g O f, x). The classifying space B(C sD) is a model for* * the homotopy colimit of D. To relate B(C sD) to our earlier discussion we define a functor D0 from C to s* *paces as follows. For a 2 ob(C ) let C #a be the category of C -objects over a, [24, II.* *6]. Let D0(a) := B ((C #a)sD) for objects a in C , where we view D as a functor on C # a. Then D0 is cofibran* *t and the canonical map D0(a) ! D(a) is a homotopy equivalence for every a in C . Moreove* *r, B(C sD) ~=colimD0. Note in passing that if D(a) is a singleton for each a in C , then the transpor* *t category C sD is identified with C and so hocolimD = BC . To make a homotopy colimit, we need a pair (C , D) consisting of a small catego* *ry C and a functor D from C to spaces. By a morphism from one such pair (C s, Ds) to an* *other, (C t, Dt), we understand a pair (F, ) consisting of a functor F :C s! C tand a* * natural transformation from Ds to DtF . 87 Remark D.1.2 Such a morphism induces a map (F, )* from hocolimDs to hocoli* *mDt. Suppose that (F0, 0) and (F1, 1) are morphisms from (C s, Ds) to (C t, Dt). L* *et ` be a natural transformation from F0 to F1 such that 1 = Dt(`) O 0. Remark D.1.3 Such a ` induces a homotopy `* from (F0, 0)* to (F1, 1)*. Proof Let I = {0, 1}, viewed as an ordered set with the usual order and then* * as a category. Then BI ~= [0, 1]. Let p: C x I ! C be the projection. The data (F0,* * 0), (F1, 1) and ` taken together define a morphism from (C sx I , Ds O p) to (C t,* * Dt). By remark D.1.2, this leads to a map from hocolim(DsOp) ~=(hocolim Ds)xBI to hocol* *imDt. * * __|_| Let C be a small category and let a 7! Fa be a covariant functor from C to the * *category of sheaves on X . Define a sheaf C sF with category structure as follows. For 0* *-connected X in X , let (C sF)(X) = C sFo(X), where each Fa(X) for a 2 ob(C ) is regardedQ* *as a discrete space. For X which is not 0-connected we define (C sF)(X) = i(C s* *F)(Xi) where the Xi are the connected components of X . In section 4.1 we used the fo* *llowing notation. Definition D.1.4 In the situation above, we let hocolima Fa := fi(C sF). Lemma D.1.5 |hocolimaFa| ' hocolima|Fa|. Proof Theorem 4.1.3, proved in appendix B above, gives |hocolimaFa| ' B|C sF|* * and propositions A.2.1, A.2.3 imply B|C sF| ~= B(C s|F?| ), where |F?| denotes the * *functor a 7! |Fa| from C to spaces. * * __|_| Corollary D.1.6 Let C be a small category and let a 7! Ea and 7! E0abe covar* *iant functors from C to the category of sheaves on X . Let = { a: Ea ! E0a} be a * *natural transformation such that every a: Ea ! E0ais a weak equivalence. Then the indu* *ced map hocolim aEa ! hocolimaE0ais a weak equivalence (between sheaves on X ). * * __|_| D.2 Stratifications and homotopy colimit decompositions Here we describe a relationship between stratifications and homotopy colimit de* *composi- tions. The point which we want to make, without proving anything definite in th* *at direction, is that a stratification of a space often comes from a homotopy colimit decompo* *sition of the space where the indexing category is an EI-category (a category in which all en* *domorphisms are isomorphisms). Compare [38]. 88 Definition D.2.1 A stratification of a topological space X is a partition of * *X into locally closed nonempty subsets, the strata, such that the closure of each stratum is a* * union of strata. The strata Xi of a stratified space X form a poset S where Xi Xj if the closur* *e of Xi contains Xj. (This is the reverse of the obvious ordering.) The tautological ma* *p X ! S does of course completely describe the stratification. Hence a stratification o* *f X can always be described by a stratification function, a map from X to a poset S . (We will* * not discuss the question which maps from X to a poset give rise to stratifications.) Definition D.2.2 A stratification of a CW-space X is a CW-stratification if t* *he closure of each stratum is a CW-subspace. Definition D.2.3 Let C be a small EI-category. Let '(C ) be the poset of iso* *morphism classes of objects in C , ordered in such a way that [C0] [C1] iff there exis* *ts a morphism C0 ! C1. Define f :BC -! '(C) in such a way that f(x) = [Ck] if the unique open cell of BC containing x corre* *sponds to a k-simplex of the form C0 C1 . . .Ck. Then f is the stratification functio* *n for a CW-stratification of BC . The following example of an EI-category is closely related to the category K i* *n section 5. Definition D.2.4 We make an EI-category J as follows. The objects are the f* *inite subsets of a fixed universe. A morphism from S1 to S2 consists of an injection * *k :S1 ! S2 and a sign function " from S2r im(f) to {-1, +1}. The composition of (k1, "1): * *S1 ! S2 and (k2, "2): S2 ! S3 is (k2k1, "3) where "3 agrees with "2 outside k2(S2) and * *with "1 O k2-1 on k2(S2 r k1(S1)). Then BJ is stratified as above. Remark D.2.5 We have BJ = 1 S1+1 by [35, 3.2]. Definition D.2.6 Let D be a contravariant functor from a small EI-category C * *to spaces (i.e., a covariant functor from C opto spaces). Then we have the projection map hocolim D -! BC and so we get a stratification on hocolim D, the pullback of the stratification* * of BC just defined. (It is true but not completely trivial that this does give a str* *atification on hocolim D.) The homotopy theoretic relationship between the values of D and the strata of h* *ocolimD is as follows. For an object C in C , the stratum with label [C] has the homoto* *py type of a homotopy orbit space D(C)hA where A is the automorphism group of C in C . 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