On the chain-level intersection pairing for PL manifolds. J.E. McClure* Department of Mathematics, Purdue University 150 N. University Street West Lafayette, IN 47907-2067 1 Introduction. Let M be a compact oriented PL manifold. The chain-level intersection pairing w* *as intro- duced by Lefschetz in [9] as a tool for constructing the intersection pairing o* *n the homology of M. The chain-level pairing is a basic ingredient in Chas and Sullivan's cons* *truction [2] of a Batalin-Vilkovisky structure on the homology of the free loop space of M. For a complete understanding of the construction in [2], it seems helpful to* * have the following theorem. Let C*M be the PL chain complex of M (see Section 3 for the * *definition). Let us say that a subcomplex of a chain complex is full if the inclusion map is* * a quasi- isomorphism. Theorem 1.1. The domain of the chain-level intersection pairing is a full subco* *mplex of C*M C*M. It might seem at first that something like Theorem 1.1 would have been neede* *d already by Lefschetz to define the intersection pairing on homology, but for that purpo* *se two weaker facts suffice: (i) For any cycles C and D in C*M, C D is homologous to an element in the dom* *ain of the intersection pairing. (ii) If C0, D0are two other cycles with C0 D0homologous to C D, then C0 D0-* * C D is the boundary of an element in the domain of the intersection pairing. (This * *is needed to show that the intersection pairing on homology is well-defined.) Theorem 1.1 is harder to prove than (i) and (ii) becauseP(among other reason* *s) a cycle in C*M C*M cannot in general be written in the form Ci Diwith Ciand Dicycl* *es. One goal of this paper is to prove Theorem 1.1 and, more generally, the anal* *ogous state- ment for the k-fold iterate of the intersection pairing; see Proposition 12.2 a* *nd Remark 12.3. It seems useful to go farther and to show that the intersection pairing give* *s C*M a structure of "partially defined commutative DGA;" this is the second (and main)* * goal of ________________________________ *Partially supported by NSF. 1 this paper (see Theorem 12.1). A consequence of this (to be proved in [13]) is * *that C*M is canonically quasi-isomorphic to an E1 chain algebra. The third goal of this paper is to give a new treatment of the chain-level i* *ntersection pairing, based on the account in [5] but with some improvements (in particular,* * the version given here avoids the need for the rather complicated arguments in [5, Section * *7] by making fuller use of the technical felicities of [3]). The results of this paper will be applied in [13] to prove two theorems abou* *t the Chas- Sullivan operations. Let S* denote the singular chain functor and let F be the * *framed little 2-disks operad [4]. Theorem A. The Batalin-Vilkovisky structure on the homology of LM is induced by* * a natural action of an operad quasi-isomorphic to S*F on a chain complex quasi-is* *omorphic to S*(LM). (Theorem A is the analog for H*(LM) of Deligne's Hochschild cohomology conje* *cture [12, Section I.1.19].) Theorem B. The Eilenberg-Moore spectral sequence converging to the homology of * *LM is a spectral sequence of Batalin-Vilkovisky algebras. The paper is organized as follows. Section 2 gives a brief discussion of the definitions of the chain-level int* *ersection pairing given in [9], [10] and [5], and explains why these versions of the definition a* *re not convenient as a starting-point for proving Theorem 1.1. Section 3 recalls the definition of the PL chain complex of a PL space. Section 4 recalls from [5] a method for making chain-level constructions by * *using relative homology. Section 5 constructs the backwards (Umkehr) map in relative homology induced* * by a PL map between compact oriented PL manifolds. In Section 6 a chain-level backwards* * map is deduced from this, using the method of Section 4. Section 7 recalls the definition of exterior product for PL chains. In Section 8, the chain-level intersection pairing is defined as the composi* *te of the exterior product and the chain-level backwards map induced by the diagonal map; the moti* *vation for this definition is that the intersection of two subsets of a set S is the i* *ntersection of their Cartesian product with the diagonal in S x S. Section 9 gives the formal definition of "partially defined commutative alge* *bra." I use Leinster's concept of homotopy algebra [11] for this purpose rather than the Kr* *iz-May def- inition of partial algebra [8] (but I will use the term "Leinster partial algeb* *ra" instead of "homotopy algebra," since the latter term seems excessively generic). Leinster'* *s definition seems to have all of the advantages and none of the disadvantages of the Kriz-M* *ay definition (cf. Remark 9.4(b)). Sections 10-13 give the proof that the intersection pairing and its iterates* * determine a Leinster partial commutative DGA structure on C*M. The proof uses a general-po* *sition result which is proved in Sections 14 and 15. I would like to thank Mike Mandell for his help and Mark Goresky and Clint M* *cCrory for useful correspondence. I would especially like to thank Shmuel Weinberger f* *or referring me to [5]. 2 2 The Lefschetz and Goresky-MacPherson definitions of the chain-level intersection pairing. This section is not needed logically for the rest of the paper; it is offered a* *s motivation for Sections 3-8. The reader may also find it helpful to consult Steenrod's account* * of Lefschetz's work on the intersection pairing ([16, pages 28-30]). This section uses some technical terms which will be defined in Sections 3-7. Lefschetz's first account of the chain-levelPintersectionPpairing C . D was * *in [9]. In this paper he uses the obvious definition: if C = mioeiand D = nioithen X (2.1) C . D = minjoei\ oj, where the signs are determined by the orientations of oei, oj and M. This form* *ula does not in fact give a chain unless all of the intersections oei\ oj have the same * *dimension, so some restriction on the pair (C, D) is necessary. Generically, the intersec* *tion of a p- dimensional PL subspace and a q-dimensional PL subspace has dimension p + q -* * dimM; pairs of PL subspaces with this property are said to be in general position. Le* *fschetz restricts the domain of the intersection pairing to pairs (C, D) for which all of the pai* *rs (oei, oj) are in general position, and he interprets terms oei\ oj which are in dimension* * less than dim C + dimD - dimM as 0. In order to prove the crucial formula (2.2) @(C . D) = (@C) . D C . @D, Lefschetz imposes a further restriction on the domain of the intersection pairi* *ng: he requires that all of the pairs (@oei, oj) and (oei, @oj) should also be in general posit* *ion.1 This assumption allows him to prove equation (2.2)by working with one pair of simplices at a ti* *me and extending additively. This definition has the disadvantage that the domain of the intersection pai* *ring is not invariant under subdivision. For example, if oe and o are 1-simplices in a 2-ma* *nifold which intersect at a point in the interior of both, then the pair (oe, o) is in the d* *omain, but if we subdivide oe and o at the intersection point we obtain a pair of chains (oe0+ o* *e00, o0+ o00) which is not in the domain (because for example the pair (@oe0, o0) is not in general* * position).2 This phenomenon is general: if (C, D) is in the domain of Lefschetz's intersect* *ion pairing with C . D 6= 0 then (unless C and D are both in dimension dim M) there will al* *ways be a subdivision in which the pair of chains determined by C and D is not in the dom* *ain. Since the PL chain complex C*M is defined to be a direct limit over subdivis* *ions (see Section 3) it would be difficult to create a partially defined operation on C*M* * C*M using the definition in [9]. This would clearly be a significant obstacle to proving (or * *even formulating) Theorem 1.1. ________________________________ 1If C and D are chains on the same triangulation and are not both in dimensi* *on dimM, this condition forces C . D to be 0, because oei and oj will intersect along a common face and* * therefore oei\ oj will be contained in @oei\ oj. 2Note also that, if the intersection point is P, then formula (2.1)gives oe.* *o = P but (oe0+oe00).(o0+o00) = 4P. 3 Lefschetz returned to the chain-level intersection pairing in [10, Section I* *V.6]. He gave a formula more general than (2.1)(equation (46) on page 212) in which the coeff* *icients are "looping coefficients" [10, Section IV.5]. This allowed him to enlarge the dom* *ain of the intersection pairing as follows: if we write supp(C) for the union of the simpl* *ices that occur in C, then C . D is defined when the three pairs (supp(C), supp(D)), (supp(@C),* * supp(D)), (supp(C), supp(@D)) are in general position; note that this condition is invari* *ant under subdivision. The "looping coefficients" used in Lefschetz's second definition are tricky * *to define ex- plicitly (see [10, Subsection 58 on page 216]). The theory has been worked out * *carefully in [7] (which I have not had an opportunity to consult) and seems to be rather com* *plicated (see the Math Review: 41 # 2663). The chain-level intersection pairing became temporarily obsolete when the cu* *p product was discovered and it was noticed that the intersection pairing in homology cou* *ld be defined using only Poincar'e duality and the cup product, without any recourse to the c* *hain level. Goresky and MacPherson returned to the chain-level intersection pairing as a* * tool for constructing an intersection pairing in intersection homology ([5, Section 2]).* * They gave an elegant construction in which the procedure of the previous paragraph is rev* *ersed: the chain-level intersection pairing is derived from the relative versions of Poinc* *ar'e duality and the cup product. Their version of the chain-level intersection pairing has the * *same domain as Lefschetz's second definition and is probably equivalent to it. Note that, with either the Lefschetz or Goresky-MacPherson definitions, the * *intersection pairing is defined on a subset of C*M x C*M rather than C*M C*M. It would be * *natural to define it on a subset of C*M C*M by letting the domain be all elements tha* *t can be written in the form X Ci Di with every pair (supp(Ci), supp(Di)), (supp(@Ci), supp(Di)), (supp(Ci), supp(@D* *i)) inPgen- eral position. For such an element, one could define the intersection pairing t* *o be Ci. Di. But this raises two issues: it is hard to tell when an element can be written i* *n the required form, and it would be hard to prove that two different ways of writing an eleme* *nt in this form lead to the same value for the intersection pairing. The definition to be given in Section 8 resolves both of these issues by def* *ining the intersection pairing (up to a dimension shift) as the composite of the exterior* * product " : C*M C*M ! C*(M x M) (see Section 7) and the chain-level backwards map !: C* (M x M) ! C*M induced by the diagonal (see Section 6); here C* (M x M) denotes the set of cha* *ins E in C*(M x M) for which both E and @E are in general position with respect to the d* *iagonal. With this definition, the domain of the intersection pairing (up to a dimension* * shift) is "-1(C* (M x M)). The analog of equation (2.2)is immediate from the fact that " and !are chain m* *aps. 4 3 PL chains. We begin by reviewing some basic definitions. A simplicial complex K is a set of simplices in Rn (for some n) with two pro* *perties: every face of a simplex in K is in K and the intersection of two simplices in K is a * *common face. (A face of a simplex oe is the simplex spanned by some subset of the vertices o* *f oe.) The simplicial chain complex of K, denoted c*K, is defined by letting cpK be* * generated by pairs (oe, o), where oe is a p-simplex of K and o is an orientation of oe, s* *ubject to the relation (oe, o) = -(oe, -o) where -o denotes the opposite orientation. We lea* *ve it as an exercise to formulate the definition of the boundary map @ (or see [15, page 15* *9]). If we choose orientations for the simplices of K (with no requirement of consistency * *among theP orientations) then every nonzero element c of c*K can be written uniquely in th* *e form nioei with all ni6= 0. The realization of K, denoted |K|, is the union of the simplices of K. A subdivision of K is a simplicial complex L with two properties: |L| = |K| * *and every simplex of L is contained in a simplex of K. The subdivision category of K has an object for each subdivision L of K and * *a morphism L ! L0whenever L0is a subdivision of L. If L0 isPa subdivision of L there is an induced monomorphism c*L ! c*L0 whic* *h takes (oe, o) to (o, oo), where the sum runs over all o 2 L0 which are contained in* * oe and have the same dimension as oe, and oo is the orientation induced by o. This makes c** * a covariant functor on the subdivision category of K. A subspace X of Rn will be called a PL space if there is a simplicial comple* *x K with X = |K|. K will be called a triangulation of X; note that X determines K up to * *subdivision by [1, page 222]. The PL chain complex of a PL space |K|, denoted C*|K|, is the direct limit colimc*L L taken over the subdivision category of K. Remark 3.1. This definition is taken from [5, Subsection 1.2], which seems to b* *e the first place where the PL chain complex was defined. Note that the direct system defining C*|K| is a rather simple one: the subdi* *vision cat- egory is a directed set (because any two subdivisions have a common refinement * *[1, page 222]), and all of the maps c*L ! c*L0are monomorphisms. It follows that each of* * the maps c*L ! C*|K| is a monomorphism. Remark 3.2. The homology of c*L is canonically isomorphic to the singular homol* *ogy of |K| by [15, Theorems 4.3.8 and 4.4.2]; since homology commutes with colimits over d* *irected sets, the homology of C*|K| is also canonically isomorphic to the singular homology o* *f |K|. Now let C be a nonzero element of C*|K|. There is a subdivision L of KPwith * *C 2 c*L, so (after choosing orientations for the simplices in L) we can write C = nioe* *iwhere the oei areSsimplices in L and the niare nonzero. We define the support of C, denoted s* *upp(C), to be oei; this is independent of the choice of L. The support of 0 is defined t* *o be the empty set. 5 4 A useful lemma. Let K be a simplicial complex. A subcomplex of K is a subset K0 of K with the p* *roperty that every face of every simplex in K0 is also in K0. A PL subspace of |K| is a space of the form |L| where L is a subcomplex of a* * subdivision of K. The next lemma is taken from Section 1.2 of [5]; it gives a way of using rel* *ative homology to make chain-level constructions. Lemma 4.1. Let K be a simplicial complex and let A and B be PL subspaces of |K|* * such that B A and dim(B) = dim(A) - 1. Let p = dim(A). (a) There is a natural isomorphism ffA,B from Hp(A, B) to the abelian group { C 2 Cp(K) : supp(C) A and supp(@C) B }. (b) The diagram ffA,B Hp(A, B) ____//_{ C 2 Cp(K) : supp(C) A and supp(@C) B } @|| |@| fflffl|ffB,; fflffl| Hp-1(B, ;)______//{ D 2 Cp-1(K) : supp(D) B and @D = 0 } commutes. Proof. For part (a), note that Hp(A, B) is isomorphic to the p-th homology of t* *he complex C*A=C*B, and this in turn is isomorphic to the quotient of the relative cycles * *by the relative boundaries. The module of relative cycles is the set S of the lemma and the mo* *dule of relative boundaries is @(Cp+1A) + CpB, which is zero because of the hypotheses.* * Part (b) is * * __ immediate from the definitions. * * |__| 5 A backwards map in relative homology. A PL map from |K| to |K0| is a continuous function f with the property that, fo* *r some subdivision L of K, the restriction of f to each simplex of L is an affine map * *with image in a simplex of K0. A PL homeomorphism is a PL map which is a homeomorphism. An m-dimensional PL manifold is a PL space M with the property that each poi* *nt of M is contained in the interior of a PL subspace which is PL homeomorphic to the m* * simplex. Let M be a compact oriented m-dimensional PL and let A and B be PL subspaces* * of M with B A. Let N be a compact oriented PL manifold of dimension n and let f : * *N ! M be a PL map. Let A0= f-1(A) and B0= f-1(B). We want to construct a map (5.1) f!: H*(A, B) ! H*+n-m(A0, B0) (one should think of this as taking a homology class to its inverse image with * *respect to f). First we need a lemma, which will be proved at the end of this section. 6 Lemma 5.1. Let X be a PL subspace of M, let X0 = f-1(X), and let W 0be an open neighborhood of X0. Then there is an open neighborhood W of X with f-1(W ) W * *0. Now let (U0, V 0) be an open pair in N with A0 U0 and B0 V 0. Using the le* *mma, we choose an open pair (U, V ) in M with A U, B V , f-1(U) U0, and f-1(V ) * * V 0. Consider the composite f* m-* 0 0 0* * 0 H*(A, B) ! H*(U, V ) ~=~Hm-*(M - U, M - V ) ! H~ (N - U , N - V ) ~=H*+n-m(U * *, V ), where the second and fourth arrows are Poincar'e-Lefschetz duality isomorphisms* * ([3, Propo- sition VIII.7.2]). By the naturality of the cap product ([3, VIII.7.6]) this c* *omposite is independent of the choice of (U, V ) and is natural with respect to (U0, V 0). * *We therefore get a map H*(A, B) ! limH*+n-m(U0, V 0) where the inverse limit is taken over all open pairs (U0, V 0) (A0, B0). This* * inverse limit is isomorphic to H*+n-m(A0, B0) by [3, Exercise 4 at the end of Section VIII.13]; * *here we use the fact that the realization of a simplicial complex is an ENR (see for example [3* *, Proposition IV.8.12]). This completes the construction of the map (5.1). For use in the next section, we need: Lemma 5.2. The diagram f! 0 0 H*(A, B) ____//H*+n-m(A , B ) @|| @|| fflffl|f! fflffl| H*-1(B) _____//H*+n-m-1(B0) commutes. * * __ Proof. This follows easily from [3, VII.12.22]. * * |__| Proof of Lemma 5.1. Fix triangulations of M and N. By subdividing the triangula* *tion of N, we can ensure that every simplex that intersects X0 is contained in W 0. * *By a further subdivision in both M and N, we can ensure that the map f is simplicial (see [6* *, top of page 15 and Lemma 1.10]). Since f is simplicial it maps simplices onto simplices, an* *d any simplex in N that maps onto a simplex which intersects X must intersect X0 and must the* *refore be contained in W 0. Now let "Xbe the union of all simplices that intersect X and * *let W be the * * __ interior of "X. Then W is an open neighborhood of X and f-1(W ) is contained in* * W 0. |__| 6 A backwards map at the chain level. Let M, N and f : N ! M be as in the previous section. We say that a PL subspace A of M is in general position with respect to f if dim(f-1(A)) dim(A) + n - m. (The dimension of the empty set is defined to be -1, so if f-1(A) is empty then* * A is in general position.) 7 Remark 6.1. For later use we make two observations. (a) Suppose that f is a composite gh, that A is in general position with res* *pect to g, and that g-1(A) is in general position with respect to h. Then A is in general * *position with respect to f. (b) Suppose that N is a Cartesian product M x M1 and f : N ! M is the projec* *tion. Then every A is in general position with respect to f. A p-chain C in C*M is said to be in general position with respect to f if dim(f-1(supp(C))) p + n - m. Let Cf*M be the set of all chains C 2 C*M for which both C and @C are in gen* *eral position with respect to f. Note that Cf*M is a subcomplex of C*M. We want to construct a chain map f!: Cf*M ! C*+n-mN. So let C 2 CfqM. Let [C] be the homology class of C in Hq(supp(C), supp(@C))* *. Let T be the abelian group { D 2 Cq+n-mN | supp(D) f-1(supp(C)) and supp(@D) f-1(supp(@C)) }. We define f!(C) to be the image of [C] under the following composite: f! -1 -1 Hq(supp(C), supp(@C)) ! Hq+n-m(f (supp(C)), f (supp(@C))) ~=T ,! Cq+n-mN Here the first map was constructed in Section 5 and the isomorphism is from Lem* *ma 4.1 (which applies because of the hypothesis that both C and @C are in general posi* *tion with respect to f). f!is a chain map by Lemmas 4.1(b) and 5.2. Remark 6.2. Note that, by the definition of T , we have supp(f!(C)) f-1(supp(* *C)). 7 The exterior product for PL chains. Let oe1 and oe2 be simplices. It is easy to see that oe1 x oe2 is a PL space; t* *hat is, there is a simplicial complex J with |J| = oe1 x oe2. Note that there is no canonical way * *to choose J, but that any two choices of J have a common subdivision. It follows that the product of any two PL spaces is a PL space. Let |K1| and |K2| be PL spaces. We want to construct a map (7.1) " : C*|K1| C*|K2| ! C*(|K1| x |K2|), called the exterior product. As a first step, let L1 and L2 be subdivisions of K1 and K2 respectively. We* * define a map (7.2) "0: c*L1 c*L2 ! C*(|K1| x |K2|) (see Section 3 for the definition of c*). 8 It suffices to define "0on generators, so for i = 1, 2 let oeibe a simplex o* *f Liwith orientation oi. Let J be a simplicial complex with |J| = oe1 x oe2. Then "0((oe1, o1) (oe2, o2)) is defined to be X (o, oo) where o runs through the simplices of J with dimension dim(oe1) + dim(oe2), and* * oo is the orientation of o induced by o1 x o2. The maps "0are consistent as L1 and L2 vary; passage to colimits gives the m* *ap ". Remark 7.1. (a) It is easy to check that " is a monomorphism. (b) The quasi-isomorphism relating c* to singular chains ([15, Theorems 4.3.* *8 and 4.4.2]) takes " to the Eilenberg-MacLane shuffle product ([3, VI.12.26.2]). Since the * *latter is a quasi-isomorphism, so is ". (c) For singular chains, the shuffle product has an explicit natural homotop* *y inverse, namely the Alexander-Whitney map ([3, VI.12.26.2]). Unfortunately the Alexander* *-Whitney map is not compatible with subdivision, so it seems to have no analog for PL ch* *ains. 8 The chain-level intersection pairing. We now have the ingredients needed to define the chain-level intersection pairi* *ng. Let M be a compact oriented PL manifold of dimension m and let : M ! M x M be the diagonal map. As in Section 6, let C* (M x M) be the subcomplex of C*(M * *x M) consisting of chains C for which both C and @C are in general position with res* *pect to . It is convenient to shift degrees so that the intersection pairing preserves* * degree. For a chain complex C* and an integer n, we will write nC* for the n-fold suspension* * of C*, that is, the chain complex with Ciin degree i + n. Let us define G2 -2m(C*M C*M) to be -2m("-1(C* (M x M)), where " is the exterior product (the G stands for "* *general position" and the subscript 2 will be explained in Section 10). The chain-level intersection pairing ~ is the composite ! -m G2 "-! -2mC* (M x M) --! C*M. Remark 8.1. It is not difficult to check that, if C and D are chains for which * *the Goresky- MacPherson intersection pairing C \ D is defined (see [5, pages 141-142]), then* * (up to the dimension shifts in the definitions of G2 and ~) C D is in G2 and ~(C D) = * *C \ D. 9 9 Leinster partial commutative DGA's. Our main goal in the rest of the paper is to show that the chain-level intersec* *tion pairing and its iterates determine a partially defined commutative DGA structure on -m* * C*M. First we need a precise definition of "partially defined commutative algebra* *." We will use the definition given by Leinster in [11, Section 2.2] (but note that Leinst* *er uses the term "homotopy algebra" instead of "partial algebra"). Let be the category of finite sets (including the empty set) and let Ch be* * the category of chain complexes. Given a functor A defined on , we will write AS (instead of A(S)) for the v* *alue of A at S. Definition 9.1. A Leinster partial commutative DGA is a functor A from to Ch * *together with chain maps ,S,T: AS `T ! AS AT for each S, T and ,; : A; ! Z (where Z is considered as a chain complex concentrated in degree 0) such that (i) the collection ,S,Tis a natural transformation of functors from x to* * Ch, (ii) the diagram ,S `T,U AS `T `U ______//_AS `T AU ,S,T `U|| |,S,T|1 fflffl|1 ,T,U fflffl| AS AT `U ____//AS AT AU commutes for all S, T, U, (iii) the diagram ,S,T AS `T ____//AS AT ~=|| |~=| fflffl|,T,Sfflffl| AT `S ____//AT AS commutes for all S, (iv) the diagram ,;,S AS ____//IIA; AS III ,| 1 ~=II$$II;fflffl|| Z AS commutes for all S, and (v) ,; and each ,S,Tare quasi-isomorphisms. 10 Remark 9.2. Definition 9.1 is the precise analog, for the category Ch, of Segal* *'s -spaces [14]. This is not immediately obvious, since a -space is a functor on the cate* *gory F of based finite sets; the point is that the maps ,S,Tin Definition 9.1 encode the same i* *nformation as the projection maps in Segal's definition. __ __ Notation 9.3. For k 1 let k denote the set {1, . .,.k}. Let 0be the empty set* *. For k 0 let Ak denote A_k. Note that a_functor A on is entirely determined by its restriction to the * *full subcategory with objects k, k 0. Remark 9.4. (a) An ordinary commutative DGA B determines a Leinster partial com* *muta- tive DGA with Ak = B k. (b) Conversely, it will be shown in [13] that Leinster partial commutative D* *GA's can be functorially replaced by quasi-isomorphic E1 DGA's. This is one reason for usin* *g Definition 9.1 instead of the Kriz-May definition of partial algebra [8, Definition II.2.4* *]; Kriz and May prove an analogous result for partial commutative simplicial algebras (in their* * sense), but they explain [8, bottom of page 35 and top of page 51] that their method of pro* *of does not work for partial commutative differential graded algebras. 10 The functor G. As a first step in showing that the intersection pairing on -m C*M extends to * *a Leinster partial commutative DGA structure, we define a functor G from to Ch with G1 =* * -m C*M (see Notation 9.3). The G stands for "general position." G2 has already been defined in Section 8. In order to define Gk for k 3 w* *e need a definition.__ __ Let R : k! k0be any map. Define 0 k R* : Mk ! M to be the composite 0 __ __ k Mk = Map (k0, M) ! Map (k, M) = M where the second arrow is induced by R. Thus the projection of R*(x1, . .,.xk0)* * on the i-th factor is xR(i).__ If R : k i k0is a surjection then we think of R* as a generalized diagonal m* *ap. For example, if k0is 1 and R is the constant map then R* : M ! Mk is the usual diag* *onal map. Let "k denote the k-fold exterior product (C*M) k ,! C*(Mk). Definition 10.1. Define G0 to be Z and G1 to be -m C*M. For k 2 define Gk to* * be the subcomplex of -mk ((C*M) k) consisting of the elements -mk C for which both "* *k(C) and "k(@C) are in general position with respect to all generalized diagonal maps, t* *hat is, " " * Gk = -mk ("-1k(CR*Mk)). k0 j the set R-1(R(i)) has only one element. (ii) For 0 j k define Gjkto be the subcomplex of -mk (C*M) k consisting* * of the chains C for which both "k(C) and "k(@C) are in general position with respect t* *o R* for all R 2 j. Thus we have a filtration Gk = Gkk Gk-1k . . .G0k= -mk (C*M) k Proposition 12.2 follows immediately from: Proposition 13.4. For each 1 j k the inclusion Gjk Gj-1kis a quasi-isomorp* *hism. For this we need a lemma which will be proved in Sections 14 and 15. Lemma 13.5. Suppose that D 2 mkGj-1kand @D 2 mkGjk. Then there is a j-th fact* *or homotopy h : Mk x I ! Mk such that hOi0 is the identity and the chains (hOi1)*("kD), (hOi1)*("k(@D)) and* * h*("(@D ')) are in general position with respect to R* for all R 2 j. Proof of Proposition 13.4. We have to show two things: (i) If D is a cycle in mkGj-1kthen there is a cycle C in mkGjkhomologous t* *o D. (ii) If C is a cycle in mkGjkwhich is the boundary of an element of mkGj-1* *kthen C is the boundary of an element of mkGjk. To show (i), choose a homotopy h as in Lemma 13.5. Then (h O i1)*("kD) is in* * the image of "k by Lemma 13.2, so we may define C = "-1k((h O i1)*("kD)). 16 C is a cycle, and the general-position property given in Lemma 13.5 implies tha* *t C is in mkGjk. Let ~, ~ 2 C0I be the 0-chains associated to 0, 1 2 I; then @' = ~ - ~. Now @(h*("(D ')))= h*("(@D ')) + (-1)|D|h*("(D ~)) - (-1)|D|h*("(D ~)) = 0 + (-1)|D|(h O i1)*("kD) - (-1)|D|(h O i0)*("kD) = (-1)|D|"k(C - D). Since "k is a quasi-isomorphism, this implies that C is homologous to D. To show (ii), let D 2 mkGj-1kwith @D = C. Choose a homotopy h as in Lemma 1* *3.5. Then (h O i1)*("kD) and h*("(@D ')) are in the image of "k by Lemma 13.2, so * *we may define E1 = "-1k((h O i1)*("kD)) and E2 = "-1k(h*("(@D '))). The general-position property given in Lemma 13.5 imply that E1 and E2 are in * *mkGjk. Now "k(@E2) = (-1)|D|+1(h*("(@D ~)) - h*("(@D ~))) = (-1)|D|+1((h O i1)*("k@D) - (h O i0)*("k@D)) = (-1)|D|+1"k(@E1 - C). Since "k is a monomorphism, this implies @((-1)|D|E2 + E1) = C. * * __ * *|__| 14 Background for the proof of Lemma 13.5. In this section we collect the tools used in the proof of Lemma 13.5. First we * *have two simple facts about affine geometry which are the heart of the proof. Recall that the a* *ffine span of a subset of Rn is the smallest affine subspace containing it. Lemma 14.1. Let oe and o be simplices in Rn such that the affine span of oe [ o* * is all of Rn. Then dim(oe \ o) dim(oe) + dim(o) - n. 17 Proof. Let U (resp., V ) be the affine span of oe (resp., o). If U \ V is empty* * the statement is obvious. Otherwise we can choose a point in U \V and move it to the origin by a* * translation; then U and V become ordinary subspaces which span Rn and we have dim(U \ V ) = dim(U) + dim(V ) - n, * * __ which proves the lemma. * * |__| Notation 14.2. If oe is a simplex in Rn and u is an element of Rn which is not * *in oe, the convex hull of oe and u will be denoted by . Lemma 14.3. Let oe and o be simplices in Rn. Let u be a point which is not in t* *he affine span of oe [ o. Then \ o = oe \ o. Proof. Let v 2 \ o. Since v 2 , we can write v in the form ffu +* * (1 - ff)s, with s 2 oe. If ff were nonzero we would have 1 1 - ff u = __v - _____s. ff ff Since v 2 o, this would imply that u is in the affine span of oe [ o. Therefore* * ff must be 0, * * __ so v is in oe, and hence in oe \ o, which proves the lemma. * * |__| Next we recall a well-known way of triangulating oe x I. By an ordered simpl* *ex we will mean a simplex with a total ordering of its vertices. Lemma 14.4. Let oe be an ordered simplex and let v0 < . . .< vl be the ordering* * of its vertices. For 0 i l let oe[i] oe x I be the convex hull of {(vj, 0) | j i} [ {(vj, 1) | j i}. Then (a) Each oe[i] is an (l + 1)-simplex. * * __ (b) The set L whose elements are the oe[i] and their faces is a triangulatio* *n of oe x I. |__| Remark 14.5. With the notation of Lemma 14.4, let o be the simplex spanned by v* *1, . .,.vl-1. Then oe[i] = for each i < l, and oe[l] = . Finally, we need a tool for extending PL maps and homotopies. 18 Construction 14.6. Let ae be a simplex in Rn and let u be an element of Rn whic* *h is not in ae. Let be a PL space with a PL homeomorphism ! : ! m . (i) Let f : ae ! be a PL map and let w be an element of . We can extend f* * to a PL map f~: ! by the formula ~f(ffx + (1 - ff)u) = !-1(ff!(f(x)) + (1 - ff)!(w)). (ii) Next suppose we are given an ordering of the vertices of ae; extend thi* *s to by letting u be the maximal element. Let OE : ae x I ! be a PL homotopy and let * *z and z0be elements of . We can extend OE to a PL homotopy O~E: x I ! as follows. Let l - 1 be the dimension of ae. For i < l we have [i] = by Remark 14.5. OE is already defined on ae[i], and we can extend it to by using the construction in part (i). For i = l, Remark 14.5 gives [l]= < x {0}, (u, 1)> = <, (u, 1)> OE is already defined on ae x {0}, and we can extend it to <,* * (u, 1)> by applying the construction in part (i) twice, taking (u, 0) to z and (u, 1) to z0. 15 Proof of Lemma 13.5. We will assume that j = k, since the other cases are essentially the same and t* *he notation is simpler in this case. So suppose we are given a D satisfying: Assumption 15.1. (i) D is in mkGk-1k. (ii) @D is in mkGkk. With the assumption that j = k, Lemma 13.5 specializes to: Lemma 15.2. There is a k-th variable homotopy h : Mk x I ! Mk such that h O i0 * *is the identity and the three chains (h O i1)*("kD), (h O i1)*("k(@D))_and_h*("(@D* * ')) are in general position with respect to R* for all surjections R : ki k0. 19 By the definition of PL manifold, M has a covering by (the interiors of) PL * *subspaces iwhich are PL homeomorphic to m . Choose PL homeomorphisms !i: i! m . Recall that, by the definition in Section 3, a PL space is given as a subspa* *ce of some Rn, and therefore inherits a metric. In particular, this is true for the PL manifol* *d M. Notation 15.3. The Lebesgue number of the covering { i} (with respect to the me* *tric just mentioned) will be denoted by j. Choose a triangulation K of M such that (i)each iis a union of simplices of K, (ii)each simplex of K is contained in some i, (iii)the restriction of each !ito each simplex of K in iis affine, (iv)the diameter of each simplex of K is less than j_2, and (v)D 2 (c*K) k. Notation 15.4. Let o1, . .,.or be the simplices of K. We fix orientations for o1, . .,.or (with no requirement of consistency amon* *g the choices); this allows us to think of the oj as generators of c*K. Property (v) of K implies that D can be written as a sum X (15.1) D = naoa1 . . .oak, a where a runs through multi-indices (a1, . .,.ak) 2 {1, . .,.r}k and na 2 Z. For 1 j r we define Ej by X (15.2) naoa1 . . .oak-1; a such thatak=j with this notation we can rewrite equation (15.1)as X (15.3) D = Ej oj. j Similarly, @D can be written as X @D = pb ob1 . . .obk b 20 and we define X Fj = pb ob1 . . .obk-1 b such thatbk=j which gives X (15.4) @D = Fj oj. j Now choose an ordering v1, . .,.vs for the vertices of K. Definition 15.5. For 1 p s let Ap be the union of the simplices of K whose * *vertices are in the set {v1, . .,.vp}. Let A0 be the empty set. Note that As is M. Let us denote the metric on M by d and the standard norm on Rm by || ||. Definition 15.6. (i) For each i, choose numbers fliand ffiiwith ||!i(x) - !i(y)|| flid(x, y) and d(x, y) ffii||!i(x) - !i(y)|| for all x, y 2 i(such numbers exist because !iand its inverse are PL maps). (ii) Let ~ be the greater of max ifliffiiand 1. We will construct, by induction over p with 0 p s, a PL homotopy OEp : Ap x I ! M with the following properties: (1)The restriction of OEp to Ap-1x I is OEp-1. (2)OEp O i0 is the identity. (3)For each x 2 Ap, t 2 I we have j d(OEp(x, t), x) _____ 2~s-p (see Notation 15.3 and Definitions 15.5 and 15.6). 21 (4)Let hp = 1 x OEp : Mk-1 x Ap x I ! Mk be the k-th factor homotopy induced by OEp. If oe is a simplex of K and j * *is a number such that oe oj Ap, then each of the_following_chains is in general po* *sition with respect to R* for all surjections R : ki k0: (a)(hp O i1)*("k(Ej oe)), (b)(hp O i1)*("k(Fj oe)), (c)(hp)*("(Fj oe ')). This will complete the proof of Lemma 15.2, because the homotopy hs will hav* *e the properties required by the lemma. The first step of the induction (the case p = 0) is trivial. Suppose that OE* *p-1 has been constructed. Notation 15.7. Let ss1, . .,.sst be the simplices of K which have a vertex at vp and are in Ap but not Ap-1. For* * each ssj, let aej be the face opposite vp; thus ssj = . Combining property (iv) of the triangulation K with property (3) of OEp-1 an* *d the fact that ~ 1, we see that for each j the diameter of the set ssj[ OEp-1(aejx I) is less than j. It follows that for each j we can choose a number i(j) with (15.5) ssj[ OEp-1(aejx I) i(j). If z0is any point in the intersection of the i(j)we can apply Construction 14.* *6(ii) (with z = vp and = i(j)) to extend OEp-1over each ssjxI. The resulting homotopy OEp will * *automatically satisfy properties (1) and (2) above. It only remains to enumerate the conditio* *ns that z0must satisfy in order for properties (3) and (4) to hold, and to show that there is * *a z0 satisfying these conditions. First we consider property (3). Let x 2 ssj and t 2 I. Let l be the dimensio* *n of ssj. With the notation of Lemma 14.4, we have (x, t) 2 ssj[e] for some e with 1 e l. * *There are two cases to consider: e < l and e = l. In the first case, Remark 14.5 allows us to write (x, t) as ff (y, t0) + (1 - ff) (vp, 1) 22 with 0 ff 1, y 2 aej and t02 I. By Construction 14.6(ii) we have (15.6) !i(j)(OEp(x, t)) = ff !i(j)(OEp-1(y, t0)) + (1 - ff) !i(j)(z0) and by property (iii) of the triangulation K we have (15.7) !i(j)(x) = ff !i(j)(y) + (1 - ff) !i(j)(vp). Now we have d(OEp(x, t), x)ffii(j)||!i(j)(OEp(x, t)) - !i(j)(x)|| by Definition 15.6(i) ffii(j)(ff ||!i(j)(OEp-1(y, t0)) - !i(j)(y)|| + (1 - ff) ||!i* *(j)(z0) - !i(j)(vp)||) by equations (15.6)and (15.7) ffii(j)fli(j)(ff d(OEp-1(y, t0), y) + (1 - ff) d(z0, vp)) by* * Definition 15.6(i) i j j ~ ff_______ + (1 - ff) d(z0, vp) 2~s-p+1 by Definition 15.6(ii) and property (3) of OEp-1 and this will be __j_2~s-pif d(z0, vp) __j___2~s-p+1. For the second case of property (3) we have e = l, and Remark 14.5 gives (x, t) = ff (y, 0) + (1 - ff) (vp, 1) for some y 2 ssj. Equation (15.7)is still valid, and equation (15.6)is replaced* * by (15.8) !i(j)(OEp(x, t)) = ff !i(j)(y) + (1 - ff) !i(j)(z0) Now we have d(OEp(x, t), x)ffii(j)||!i(j)(OEp(x, t)) - !i(j)(x)|| by Definition 15.6(* *i) ffii(j)(1 - ff) ||!i(j)(z0) - !i(j)(vp)|| by equations (15.* *7)and (15.8) ffii(j)fli(j)d(z0, vp) by Definition 15.6(i) ~d(z0, vp) by Definition 15.6(ii) and for this to be __j_2~s-pit again suffices to have d(z0, vp) __j___2~s-p* *+1. It remains to determine the conditions that z0must satisfy in order for OEp * *to have property (4). __ _ We begin with some general observations. Let S : k i lbe a surjection and le* *t im(S*) denote the image of S*; note that the dimension of im(S*) is lm (where m is the* * dimension of M). Since S* is 1-1, the definition of general position simplifies somewhat:* * a chain C is in general position with respect to S* if and only if dim (supp(C) \ im(S*)) dim(supp(C)) + (l - k)m. Let us observe that, if O1, . .,.Ok are simplices of M and C is "k(O1 . . .Ok* *), then supp(C) \ im(S*) is homeomorphic to the subspace Y " Oi j2_lS(i)=j 23 of Ml. Thus X i " j (15.9) dim(supp(C) \ im(S*)) = dim Oi . j2_l S(i)=j It follows that C is in general position with respect to S* if and only if X i " j X dim Oi dim(Oi) + (l - k)m. j2_l S(i)=j i Combining this with Assumption 15.1(i), Definition 13.3, and equation (15.1), w* *e have X i " j X (15.10) dim oai dim(oai) + (l - k)m j2_l S(i)=j i for all a with_na 6= 0 and all S such that S-1(k) is a single point. In particu* *lar, let P be a subset of k which doesn't_contain k and let S be any surjection which takes P* * to a point and is 1-1 on the rest of k. In this situation (15.10)simplifies to i" j X (15.11) dim oai dim(oai) - (|P | - 1)m, i2P i2P where |P | is the cardinality of P . Now let us consider_what_conditions z0must satisfy in order for part (a) of * *property (4) to be valid. Let R : k i k0be a surjection. By the inductive hypothesis, proper* *ty (4a) is already valid when oe is a simplex of Ap-1, so we may assume that oe is a simpl* *ex of Ap which is not in Ap-1 (see Definition 15.5). Denote the set R-1(R(k)) by Q. With the notation of equation (15.2), (hpO i1)*("k(Ej oe)) \ im(R*) is home* *omorphic to the union over all a such that na 6= 0 of i Y " j i " j oai x OEp(oe x {1}) \ oai . j6=R(k)R(i)=j i2Q-{k} Thus we want to choose z0so that the inequality X i " j i " j (15.12) dim oai + dim OEp(oe x {1}) \ oai j6=R(k) R(i)=j i2Q-{k} X dim(oai) + dim(OEp(oe x {1})) + (k0- * *k)m i, . .,.. T Let us write _a for the simplex i2Q-{k}oaithat occurs in property (*). In* * order for z0 to satisfy property (*), each of the pairs consisting of a simplex and a simplex !i(q)(_a) must be in general position. By Lemma 14.1, this condition is* * automatically satisfied (with no restriction on z0) by those pairs for which the affine span * *of Ol[ !i(q)(_a) is all of Rm . For the remaining pairs, it suffices by Lemma 14.3 that !i(q)(z* *0) should not be in the affine span of Ol[ !i(q)(_a) (note that Oland !i(q)(_a) are in genera* *l position by the inductive hypothesis). Since this affine span is nowhere dense, the set of * *allowable z0for each such pair contains an open dense subset of a neighborhood of vp. If we now* * let l, a and oe vary through all relevant choices, we still have an open dense subset U of a* * neighborhood of vp for which property (*), and hence property (4a), is valid. A similar argument shows that there is an open dense subset V of a neighborh* *ood of vp for which property (4b) is valid and an open dense subset W of a neighborhood o* *f vp for which property (4c) is valid. If z0 is chosen in the intersection of U, V and W* * , all parts of property (4) are valid. This concludes the proof. References [1]Bryant, J.L. Piecewise linear topology. 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