ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES
DAVID BLANC
Abstract.We define a sequence of purely algebraic invariants - namely, *
*classes in the
Quillen cohomology of the -algebra ss*X - for distinguishing between diff*
*erent homotopy
types of spaces. Another sequence of such cohomology classes allows one t*
*o decide whether a
given abstract -algebra can be realized as the homotopy -algebra of a spa*
*ce.
1.Introduction
The usual Postnikov system for a (simply-connected) CW complex X serves to det*
*ermine
its homotopy type. One begins with purely algebraic data, consisting of the hom*
*otopy groups
(ssnX)1n=2. However, in order to construct the succesive approximations X(n)(*
*n 2), with
X ' holimX(n), one must specify a sequence of cohomology classes kn 2 Hn+2(X(*
*n); ssn+1X)
(see [W , IX, x2]). These can hardly qualify as algebraic invariants, since th*
*eir description
involves the cohomology groups of topological spaces. In this paper we show th*
*at if one is
willing to invest the graded group ss*X := (ssnX)1n=1with some further algebra*
*ic structure,
the additional information needed to determine the homotopy type of X can be de*
*scribed in
purely algebraic terms.
The structure needed on ss*X is that of a -algebra - i.e., a graded group equi*
*pped with an
action of the primary homotopy operations (Whitehead products and compositions)*
*. In this
context, the additional data needed consists of cohomology classes in the Quill*
*en cohomology
of this -algebra - which can be defined as usual in algebraic terms (see x4.1*
* below). We
show:
Theorem A. Given two realizations X and X0 of a -algebra J*, there is a s*
*uccessively
defined sequence of "difference obstructions" ffin 2 Hn+1(J*; nJ*), taking valu*
*e in the Quillen
cohomology groups of J*, with coefficients in the J*-module nJ*, whose van*
*ishing implies
that X ' X0.
(See Theorems 4.18 and 4.21 below). The (n + 1)-st cohomology class is defined *
*whenever the
n-th Postnikov section of the simplicial space resolutions of the spaces X and *
*X0, respectively,
agree, up to homotopy. Even though the obstructions are defined in terms of a s*
*pecific choice
of -algebra resolution of J*, in fact they depend only on the homotopy type of *
*the Postnikov
sections.
Moreover, these cohomology groups can also be used to determine the realizabil*
*ity of an
abstract -algebra as the homotopy groups of some space:
Theorem B. Given a -algebra J*, there is a successively defined sequence of "c*
*haracteristic
classes" 2 Hn+2(J*; nJ*), which vanish if and only if J* is realizable by a to*
*pological space.
(See Theorems 4.8 and 4.15 below). The vanishing requirement should be understo*
*od in the
sense of an obstruction theory: if any such sequence of cohomology classes vani*
*shes, the -
algebra is realizable; if one reaches a non-trivial obstruction, one must back*
*-track, and try to
____________
Date: (Revised version) September 18, 1998.
1991 Mathematics Subject Classification. Primary 55S45; Secondary 55Q35, 55P1*
*5, 18G10, 18G55.
Key words and phrases. homotopy invariants, simplicial resolution, homotopy t*
*ype, ;-algebra Quillen
cohomology.
1
2 DAVID BLANC
vary the choices involved in order to obtain a realization. These choices again*
* depend only the
homotopy type of a suitable Postnikov section - this time, of a simplicial re*
*solution we are
trying to construct for the putative topological space X realizing J*. See Pr*
*oposition 4.10
below.
The theory is greatly simplified if we are only interested in the rational hom*
*otopy type of a
simply-connected space X. In that case, a rational -algebra is simply a graded *
*Lie algebra
over Q, and the cohomology theory in question reduces to the usual cohomology o*
*f Lie algebras.
Theorem A thus provides an integral version of (the dual to) the Halperin-Stash*
*eff obstruction
theory for rational homotopy types (see [HS ] and x4.22 below).
It is in order to be able to deal with this case, too (and other possible vari*
*ants - see
x2.14 below), that we have stated our results for a general model catgory C (su*
*bject to certain
somewhat restrictive simplifying assumptions on C - not all of which are really*
* necessary). For
technical convenience we have chosen to describe the ordinary topological versi*
*on of our theory
within the framework of simplicial groups, rather than topological spaces (see *
*x4.12 below).
1.1. notation and conventions. T will denote the category of topological space*
*s, and T*
that of pointed connected topological spaces with base-point preserving maps. T*
*he base-point
will be written * 2 X.
The category of groups is denoted by Gp, that of graded groups by grGp, th*
*at of (left)
R-modules by R-Mod , and that of sets by Set.
1.2. Definition. is the category of ordered sequences n = <0; 1; : :;:n> (n *
*2 N), with
order-preserving maps. opis the opposite category. As usual, a simplicial ob*
*ject over any
category C is a functor X : op! C; more explicitly, it is a sequence of objec*
*ts {Xn}1n=0in C,
equipped with face maps di: Xn ! Xn-1 and degeneracies sj : Xn ! Xn+1 (0 i; *
*j n),
satisfying the usual simplicial identities ([May , x1.1]). We usually denote *
*such a simplical
object by Xo. The category of simplicial objects over C is denoted by sC. T*
*he standard
embedding of categories c(- )o : C ! sC is defined by letting c(X)o 2 sC denote*
* the constant
simplicial object on any X 2 C (with c(X)n = X, di= sj= idX ).
The category of simplical sets will be denoted by S, rather than sSet, tha*
*t of pointed
connected simplicial sets by S*, and that of simplicial groups by G. If we con*
*sider a simplicial
object Xo over G, say, we shall sometimes call n in X1; : :;:Xn; : :t:he exte*
*rnal simplicial
dimension, written (-)extn, in distinction from the internal simplicial dimen*
*sion k, inside G,
denoted by (-)intk. In this case we shall sometimes write (Xo)intk2 sGp, in*
* contrast with
Xn 2 G, to emphasize the distinction.
The standard n simplex in S is denoted by [n], generated by oen 2 [n]n, wi*
*th k[n]
the subobject generated by dioen for i 6= k.
If we denote by the category obtained from by omitting the objects {*
*k}1k=n+1,
the category of functors ( )op! C is called the category of n-simplicial o*
*bjects over C -
written sC. If C has enough colimits, the obvious truncation functor trn: s*
*C ! sC has
a left adjoint aen : sC ! sC, and the composite skn:= aen O trn: sC ! sC *
* is called the
n-skeleton functor.
1.3. organization. In section 2 we review some background material on closed mo*
*del category
structures for categories of simplicial objects and show how certain convenient*
* CW resolutions
may be constructed therein. In section 3 we construct Postnikov systems for suc*
*h resolutions,
and define the action of the fundamental group on them; and in section 4 we exp*
*lain how these
resolutions are determined in terms of appropriate cohomology classes, which ma*
*y also be used
to determine the realizability of a (generalized) -algebra (Theorems 4.8 and 4.*
*15), as well as
to distinguish between different possible realizations (Theorems 4.18 and 4.21).
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 3
1.4. Acknowledgements.I would like to thank Dan Kan for suggesting that I conti*
*nue the
project begun in [DKS1 ] and [DKS2 ], and Bill Dwyer, Phil Hirschhorn and Emman*
*uel Dror-
Farjoun for several useful conversations. I am especially grateful to Hans Baue*
*s and Paul Goerss
for pointing out the necessity of taking into account the action of the fundame*
*ntal group in
describing the coefficients of the cohomology groups. I would also like to than*
*k the referee for
his comments.
It should be noted that Baues had previously constructed the first difference *
*obstruction of
Theorem A, lying in H2(J*; J*), by different methods, and has since extended hi*
*s construction
to the full range of invariants we define here: see [Ba3]. Yet a third descri*
*ption of these
invariants, more in the spirit of the original approach of Dwyer, Kan, and Stov*
*er, is planned in
[BG ].
2. model categories of simplicial objects
We first review some background material on model category structures for cate*
*gories of
simplicial objects, in particular a slightly expanded version of structure defi*
*ned in [DKS2 ], and
show how one can construct CW resolutions in such a context.
2.1. model categories. A model category in the sense of Quillen (see [Q1 ]) is *
*a category C
equipped with three distinguished classes of morphisms: W (weak equivalences),*
* C, and F,
satisfying the following assumptions:
(1) C has all small limits and colimits.
(2) W is a class of quasi-isomorphisms (i.e., there is some functor F : C ! *
*D such that
f 2 W , F (f) is an isomorphism).
p
(3) Any morphism f : A ! B in C has a factorization A -i!C -! B (f = p O i*
*) with
i 2 C \ W and p 2 F; moreover, this factorization is unique up to weak e*
*quivalence, in
0 p0
the sense that if A i-!C0-! B is another such factorization of f (i02 C *
*\ W, p02 F),
then there is a map h : C ! C0 such that h O i = i0 and p0O h = p.
p
(4) Similarly, any morphism f : A ! B in C has a factorization A i-!C -!B *
*(f = p O i)
with i 2 C and p 2 F \ W - again unique up to weak equivalence.
(5) We will assume here that the factorizations above may be chosen functoriall*
*y (though this
is not included in the original definition in [Q1 , I, x1]).
We call the closures under retracts of C and F the classes of cofibrations and*
* fibrations,
respectively. The definition given here is then equivalent to the original on*
*e of Quillen in
[Q1 , Q3] (see [Bl5, x2]).
An object X 2 C is called fibrant if X ! *f is a fibration, where *f is th*
*e final object
of C; similarly X is cofibrant if *i! X is a cofibration (*i= initial object*
*). If X 2 C is
cofibrant and Y 2 C is fibrant, we denote by [X; Y ]C (or simply [X; Y ]) the s*
*et of homotopy
equivalence classes [f] of maps f : X ! Y . For this to be defined we in fa*
*ct need only
require X to be cofibrant or Y to be fibrant (cf. [Q1 , I,x1]). A map in W \ F *
*is called a trivial
fibration, and one in W \ C a trivial cofibration.
Given a model category , one can "invert the weak equivalences" to*
* obtain the
associated homotopy category hoC, in which the set of morphisms from X to Y is*
* just [X; Y ]
(at least when X and Y are both fibrant and cofibrant). See [Q1 , I], [Q3 , I*
*I,x1], or [Hi, ch.
IX-XI] for some basic properties of model categories.
2.2. pointed model categories. In a pointed model category - i.*
*e., one with
a zero object, denoted by 0 or * (= *f = *i) - we may define the fiber of a m*
*ap (usually: a
f
fibration) f : X ! Y to be the pullback of X -!Y *, and the cofiber of a map*
* (usually:
a cofibration) i : A ! B to be the pushout of * A -i!B. The suspension A*
* of a
4 DAVID BLANC
(cofibrant) object A 2 C is then defined to be the cofiber of A q A ! A x I,*
* where A x I
is any cylinder object for A (cf. [Q1 , I,1,Def. 4]); it is unique up to homo*
*topy equivalence.
Similarly, the loops X of a fibrant object X is the fiber of XI ! X x X, wh*
*ere XI is a
path object for X (ibid.). Finally, the cone CA of a (cofibrant) object A 2 C*
* is the cofiber
of either map A ,! A x I. See [Q1 , I,2.8-9].
2.3. simplicial objects. For any category C with coproducts, one has a simplic*
*ial structure
(cf. [Q1 , II, x1]) on the category sC of simplicial objects over C, defined a*
*s usual by:
`
(i)For any simplicial set A 2 S and X 2 C, we define X ^A 2 sC by (X ^A)n := *
* a2AnX,
with the face and degeneracy maps induced from those of A. We denote the c*
*ofiber of
A ^*! A ^X by A ^ X. `
Now for Xo 2 sC we define Xo A 2 sC by (Xo A)n := a2AnXn (the diagonal
of the bisimplicial object Xo^A).
(ii)For any Xo; Yo 2 sC we define the function complex map (Xo; Yo) by
map (Xo; Yo)n := Hom sC(Xo [n]; Yo);
where [n] 2 S denotes the standard simplicial n-simplex.
2.4. Definition.For any complete category C, the matching object functor M : So*
*pxsC ! C,
written MAXo for a (finite) simplicial set A 2 S and any Xo 2 sC, is defined b*
*y requiring
that M[n]Xo := Xn, and if A = colimiAi then MAXo = limiMAiXo (see [DKS2 , x*
*2.1]).
This may be defined by adjointness, via:
HomsC(Z A; Xo) ~=Hom C(Z; MAXo)
for Xo 2 sC and Z 2 C.
In particular, we write MknXo for MAXo where A is the subcomplex of skn-1[n] g*
*enerated
by the last (n - k + 1) faces (dkoen; : :;:dnoen). When C = Set or Gp, f*
*or example, this
reduces to:
(2.5) MknXo = {(xk; : :;:xn) 2 (Xn-1)n+1| dixj= dj-1xi for allk i < j n};
and the map ffikn: Xn ! MknXo induced by the inclusion A ,! [n] is defined *
* ffin(x) =
(dkx; : :;:dnx). The original matching object of [BK , X,x4.5] was M0nXo = M@[*
*n]Xo, which
we shall further abbreviate to MnXo; note that each face map dk : Xn+1 ! Xn *
* factors
through ffin := ffi0n. See also x3.1 below and [Hi, XVII, 87.17].
The dual construction yields the colimit LnXo, sometimes called the "n-th`latc*
*hing object"
of Xo - see [DKS1 , x2.3(i)]. For Xo 2 x, for example, we have LnXo := 0in-1*
* Xn-1= ~,
where for any x 2 Xn-k-1 and 0 i j n - 1 we set sj1sj2: :s:jkx in the i*
*-th copy of
Xn-1 equivalent to si1si2: :s:ikx in the j-th copy of Xn-1 whenever the simplic*
*ial identity
sisj1sj2: :s:jk= sjsi1si2: :s:ik
holds (so in particular sjx 2 (Xn-1)i is equivalent to six 2 (Xn-1)j+1 for al*
*l 0 i j
n - 1). The map oen : LnXo ! Xn is defined oen(x)i= six, where (x)i2 (Xn-1)i.
There are (at least) two ways to extend a given model category structure on C *
*to sC:
2.6. Definition.In the Reedy model structure on sC (see [R ] or [Hi, XVII, x88]*
*), a simplicial
map f : Xo ! Yo is
(i)a weak equivalence if fn : Xn ! Yn is a weak equivalence in C for each n *
*0;
(ii)a (trivial) cofibration if fn q oen : Xn qLnXoLnYo ! Yn is a (trivial) cofi*
*bration in C for
each n 0;
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 5
(iii)a (trivial) fibration if fn x ffin : Xn ! Yn xMnYoMnXo is a (trivial) fib*
*ration in C for
each n 0.
Note that these definitions imply that Xo 2 sC is fibrant if and only if the m*
*aps ffin : Xn !
MnXo are fibrations (in C) for all n.
We shall require another structure, originally called the "E2-model category" *
*(see [DKS1 ,
x3] and x4.20 below), defined under the following
2.7. Assumption.Assume that is a pointed cofibrantly generated mod*
*el category,
in which every object is fibrant (this holds, for example, if C = T* or C = G*
*). Let F = FC
be a small full subcategory of C with the following properties:
(i)There is a subset {M}ff2F^ ObjF consisting of cogroup objects for C -*
* so there is
a natural group structure on Hom C(M; Y ) for any Y 2 C.
(ii)F is closed under coproducts, and`every object Z 2 F is weakly equivalent*
* to some
(possibly infinite) coproduct iM with ffi 2 F^ - so Z is a homot*
*opy cogroup
object (i.e., [Z; Y ]C has a natural group structure). However, we do no*
*t require the
morphisms in F to respect the cogroup structure, even up to homotopy.
(iii)F is closed under suspensions - that is, for each X 2 F, there is a mo*
*del for X in
F. We also assume CM 2 F for every ff 2 ^F (x2.2).
We now wish to define an algebraic model for the collection of sets of homotop*
*y classes of
maps {[X; Y ]C}X2F , for a given object Y 2 C. This is provided by the followi*
*ng
2.8. Definition.Given F C as in x2.7, we define a F-algebra to be a functor ho*
*(F)op!
Set, which takes coproducts in F to products in Set (compare [Dr]).
The category of all F-algebras will be denoted by F -Alg, and the functor [*
*hoF; -] :
C ! F -Alg defined ([B; Y ])B2hoF will be denoted by ssF. F -Alg is a category*
* of universal
graded algebras, or CUGA, in the sense of [BS , x2.1]. In particular,`the`free*
* F-algebras are
those isomorphic to ssFX for some X 2 F. If we assume that X ' ff2F^t2TffM<*
*ff>t for
some F^-graded set T*, we say that ssFX is the free F-algebra generated by T*.
If f : X ! Y is a morphism in C, the induced morphism of F-algebras, ssFf*
* : ssFX !
ssFY , will be denoted simply by f#.
2.9. Remark.Since all objects in F are homotopy equivalent to coproducts of obj*
*ects from the
set ^F, a F-algebra may be thought of more concretely as an ^F-graded group *
* - i.e., a
collection of groups (Gff)ff2F^- equipped with a (contravariant) action of the *
*homotopy classes
of morphisms in F on them, modeled on the action of such homotopy classes on {[*
*M; Y ]}ff2F^
by precomposition (cf. [W , XI, x1]).
We shall write ssffX for (ssFX)ff:= [M; X], and ssff+kX for [kM; X].
2.10. Definition.As usual, a F-algebra X is called abelian if Hom F -Alg(X; A) *
*has a natural
abelian group structure for any A 2 F -Alg (see [BS , x5.1] for an explicit de*
*scription.). In
particular, for any X 2 F -Alg, its abelianization Xab may be defined as in [B*
*S , x5.1.4] as a
suitable quotient of X. Another abelian F-algebra which may be defined for any *
*X is its loop
algebra X, defined by X(B) := X(B) (cf. [DKS2 , x9.4]; recall that F is clo*
*sed under
suspension). The fact that it is abelian follows as in [Gr, Prop. 9.9]. The (a*
*belian) category
of abelian F-algebras will be denoted by F -Algab.
2.11. Example. In C = T*, let F denote the subcategory whose objects are we*
*dges of
spheres of various dimensions; then for any space X 2 T*, the functor ssFX is d*
*etermined up
to isomorphism by ss*X, the homotopy -algebra of X - that is, its homotopy grou*
*ps, together
with the action of the primary homotopy operations (Whitehead products and comp*
*ositions)
6 DAVID BLANC
on them. See [Bl2, x2] or [St, x4]. In particular, the abelian -algebras are th*
*ose for which all
Whitehead products are trivial (cf. [Bl2, x3]).
2.12. Remark.This example does not quite fit our assumptions (x2.7), since the*
* spheres are
only co-H-spaces, i.e., homotopy cogroup objects in T*. This does not affect *
*the arguments
at this stage - in fact, this is the original example of an "E2-model catego*
*ry" in [DKS1 ].
However, for our purposes G appears to be more convenient than T* as a model *
*for the
homotopy category of (connected) spaces (see [K2 ]; also, e.g., [Bl6, x5]).
In fact, in all the examples we have in mind the objects in C will have an (un*
*derlying) group
structure, so it will be convenient to add to x2.7 the following additional
2.13. Assumption.C is equipped with a faithful forgetful functor ^U: C ! D - *
* where D is
one of the "categories of groups" D = Gp, grGp, G, R-Mod , or sR-Mod , for some*
* ring R -
and the cogroup objects M 2 ^Fof x2.7(i) are in the image of its adjoint ^F*
*, with the group
structure on Hom C(M; X) induced from that of U^(X). When D = G or D = sR*
*-Mod ,
the objects M must actually lie in the image of the composite ^FO F 0: S*
* ! C, where
F 0: S ! D is adjoint to the forgetful functor U0: D ! S.
We also assume that the adjoint pair (U^; ^F) create the model category stru*
*cture on C in
the sense of [Bl5, x4.13] - so in particular ^Ucreates all limits in C (cf. [Mc*
*1 , V,x1]).
2.14. Remark.In fact, the categories C in which shall be interested are the fol*
*lowing:
o C = G, so sC, the category of bisimplicial groups, is a model for simplicia*
*l spaces;
o C = Gp, so sC = G is a model for the homotopy category of connected topolog*
*ical spaces
of the homotopy type of a CW complex;
o C = dL, the category of differential graded Lie algebras (or equivalently,*
* C = sLie), so
sC is a model for simplicial rational spaces;
o C = Lie, the category of Lie algebras, so sLie is a model for (simply conne*
*cted) rational
spaces (cf. [Q3 , II,x4-5]);
o C = R-Mod , the category of (left) modules over a not-necessarily commutat*
*ive, possibly
graded, ring R, so sC is a model for chain complexes over R.
and it is the desire to give a unified treatment for these five cases that forc*
*es upon us the
somewhat unnatural set of assumptions we have made in x2.7 and here.
2.15. Definition.A map f : Vo ! Yo in sC is called F-free if for each n 0, th*
*ere is
a) a cofibrant object Wn which is weakly equivalent to an object in F;
b) a map 'n : Wn ! Yn in C which induces a trivial cofibration (VnqLnVoLnYo)qW*
*n ! Yn.
2.16. The resolution model category. Given a model category C and a subcategory*
* F as
in x2.7, we define the resolution model category structure on sC, with respect *
*to F by setting
a simplicial map f : Xo ! Yo to be
(i)a weak equivalence if ssFf is a weak equivalence of ^F-graded simplicial g*
*roups (x2.9).
(ii)a cofibration if it is a retract of an F-free map;
(iii)a fibration if it is a Reedy fibration (Def. 2.6(iii)) and ssFf is a (le*
*velwise) fibration of
simplicial groups (that is, for each B 2 F and each n 0, the group hom*
*omorphism
[B;fn] ext ext
[B; Xn] ---! [B; Yn] is an epimorphism (where for Go := [B; Yo] 2 G, Go *
*denotes
the connected component of the identity) - see [Q1 , II,3.8].
This was originally called the "E2-model category structure" on sC. See [DKS*
*1 , x5] for
further details.
2.17. Example. Let C = Gp with the trivial model category structure: i.e., on*
*ly isomor-
phisms are weak equivalences, and every map is both a fibration and a cofibrati*
*on. Let FGp
be the category of all free groups (which are the cogroup objects in Gp - cf.*
* [K1 ]). The
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 7
resulting resolution model category structure on G := sGp is the usual one (c*
*f. [Q1 , II, x3]).
This observation is due to Pete Bousfield. We can then iterate the process by l*
*etting FG be
the category of (coproducts of) the G-spheres, defined: Sn := F Sn-12 G - see*
* [Mi] - (with
S0 := GS0), and obtain a resolution model category structure on sG (bisimplici*
*al groups).
Note that if we tried to do the same for C = Set, there are no nontrivial co*
*group objects,
while in S not all objects are fibrant (see x2.7). The category T* of pointed *
*topological spaces,
which is the main example we actually have in mind, does not quite fit our assu*
*mptions (but
see x2.12 above).
Motivation for the name of "resolution model category" is provided by the foll*
*owing
2.18. Definition.A resolution of an object Xo 2 sC (relative to F) is a cofibra*
*nt replacement
for Xo in the resolution model category on sC determined by F: that is, it i*
*s any cofibrant
object Qo, equipped with a weak equivalence to Xo, which may be obtained fr*
*om the
factorization of * ! Xo as * cof-!Qo fib+w.e.----!Xo - and is thus unique up to*
* weak equivalence,
by x2.1(4)).
More classically, a (simplicial) resolution for an object X 2 C is a resolutio*
*n of the constant
simplicial object c(X)o (cf. x1.2) in sC.
2.19. Functorial resolutions. The construction of [St, x2] provides canonical r*
*esolutions in
sC, defined as follows: consider the comonad L : C ! C given by
a a [ a a
(2.20) LY = MOE CM ;
ff2F^OE2HomC(M;Y ) ff2F^2HomC(CM;Y )
by which we mean the the coproduct, over all OE : M ! Y , of the colimits*
* of the various
diagrams consisting of an inclusion MOE! CM for each : CM ! Y s*
*uch that
|M= OE. The counit " : LY ! Y is "evaluation of indices", and the comult*
*iplication
# : LY ,! L2Y is the obvious "tautological" one. Note that LY 2 F for any Y 2*
* C by our
assumptions on F (x2.7).
Given X 2 C, we define its canonical resolution Qo ! X by Qn := Ln+1X, w*
*ith the
degeneracies and face maps induced as usual by " and # (see [Gd , App., x3]).
The construction can be modified so as to yield resolutions for arbitrary Yo *
*2 sC, and not
only c(X)o. Moreover, it has the advantage that ssFg : ssFQn ! ssFYn is cle*
*arly surjective
for all n, so g can be changed into a fibration (Def. 2.16(iii)) by simply chan*
*ging each Qn up
to homotopy, which yields the factorization needed for x2.1(3).
An alternative (noncanonical) construction of a resolution is given in Proposi*
*tion 2.41 below.
2.21. representing objects for sC. Just as the spheres "represent" the weak equ*
*ivalences in
the usual model structure on T*, for example, in the sense that a map f : X ! Y*
* is a weak
equivalence if and only if it induces an isomorphism f* : [Sn; X] ! [Sn; Y ] fo*
*r each n 0, we
may similarly define representing objects for the resolution model category (co*
*mpare [DKS2 ,
x5.1]):
2.22. Definition.Given a model category C and a subcategory F as above, for eac*
*h n 0,
the n-dimensional simplicial F-sphere, denoted by SnF, is the subcategory n*
*F of sC,
whose objects are of the form nX := X ^ Sn for X 2 F, where Sn = [n]= _[n]*
* is the
usual simplicial n-sphere (see x2.3(i)).
Note that each such nX is cofibrant (in fact, free) in the resolution model *
*category sC.
Moreover, by the definition of the simplicial structure on sC (x2.3), nX is*
* also a cogroup
object in sC.
8 DAVID BLANC
Given Yo 2 sC, choose some fibrant replacement Xo (that is, factor Yo ! * as Y*
*o cof+w.e.----!
Xo fib-!*, using x2.1(3)) and define ^ssnYo(also written [SnF; Yo]) to be *
*the F^-graded set
ss0map (SnF; Xo). This definition is independent of the choice of Xo.
We define a map f : Xo ! Yo in sC to be an F-equivalence if it induces isomo*
*rphisms in
^ssn(-)for all n 0.
2.23. fibration sequences. Let F C be as in x2.7, and Xo ! Yo a fibration*
* in the
resolution model category sC (x2.16), with fiber Fo (x2.2). Then as usual we *
*have the long
exact sequence of the fibration:
(2.24) . .!.^ssn+1Yo@*-!^ssnFo! ^ssnXo! ^ssnYo! . .!.^ss0Yo;
(see [Q1 , I,3.8]), which in fact may be constructed in this case as for S* (se*
*e [May , 7.6]).
2.25. Definition.Given Xo 2 sC, , we define the n-cycles object of Xo, written *
*ZnXo, to
be the fiber of ffin : Xn ! MnXo (see x2.4), so ZnXo = {x 2 Xn| dix = 0 fori *
*= 0; : :;:n}
(cf. [Q1 , I,x2]). Of course, this definition really makes sense only when ff*
*in is a fibration
in C. Similarly, the n-chains object of Xo, written CnXo, is defined to be*
* the fiber of
ffi1n: Xn ! M1nXo.
Note that for any W 2 C and fibrant Xo 2 sC we have natural adjunction iso*
*morphisms
Hom sC(W ^ Sn; Xo) ~=Hom C(W; ZnXo) and HomsC(W ^ Dn; Xo) ~=Hom C(W; CnXo) (*
*where
Dn := n=0[n] 2 S is a simplicial model for the n-disc).
If Xo is fibrant, the map d0 = dn0:= d0|CnXo: CnXo ! Zn-1Xo is the pullbac*
*k of
ffin : Xn ! MnXo along the inclusion : Zn-1Xo ! MnXo (where (z) = (z; 0; : :*
*;:0)), so
d0 is a fibration (in C), fitting into a fibration sequence
jXn d0
(2.26) . .Z.n-1Xo ! ZnXo -! CnXo -! Zn-1Xo
(see [DKS2 , Prop. 5.7]). Moreover, there is an exact sequence of F-algebras
(d0)# q
(2.27) ssFCn+1Xo ---! ssFZnXo -!^ssnXo! 0;
(see [DKS2 , Prop. 5.8]), which provides a (relatively) explicit way to recover*
* s^snXofrom Xo.
Finally, the composition of the boundary map @* : Zn-1Xo ! ZnXo of the fibra*
*tion
sequence (2.26)with d0 is trivial, so by (2.27)it induces a map of F-algebras*
* from
^ssn-1Xo~=^ssn-1Xo(x2.10) to ssFZnXo which, composed with the map q in (2.27)*
*, de-
fines a "shift map" s : ^ssn-1Xo! ^ssnXo(see [DKS2 , Prop. 6.2]).
2.28. the simplicial F-algebra. Applying the functor ssF dimensionwise to any *
*simplicial
object Xo 2 sC yields a simplicial F-algebra Go = ssFXo, which is in particular*
* an ^F-graded
simplicial group; its homotopy groups form a sequence of ^F-graded groups which*
* we denote by
(ssnssFXo)1n=0, and each ssnssFXo is a F-algebra.
Note that as for any (graded) simplicial group, the homotopy groups of Go may *
*be computed
using the Moore chains C*Go, defined CnGo := \ni=0Ker{di: Gn ! Gn-1} (cf. x*
*2.25 and
[May , 17.3]), and we have the following version of [Bl8, Prop. 2.11]
2.29. Lemma. For any fibrant Xo 2 sC, the inclusion : CnXo ,! Xn induces a*
*n isomor-
phism ? : ss*CnXo ~=Cn(ss*Xo) for each n 0.
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 9
Proof.(a) First, note that any trivial cofibration j : A ,! B in S induces *
*a fibration
j* : MBXo ! MAXo in C.
To see this, by assumption 2.13 it suffices to consider C = D, (since by [Bl*
*5, Def. 4.13], f
is a fibration in C if and only if Uf is a fibration in D), and in fact the o*
*nly nontrivial case
is when D = G (where the fibrations are maps which surject onto the identity *
*component
- see [Q1 , II, 3.8]). Note that in internal simplicial dimension k we have *
* (MAXo)intk~=
Hom sGp(F A; (Xo)intk) (see x1.2) for A 2 S, where F denotes the (dimensionwis*
*e) free group
functor. Since F A is fibrant in sGp, F j : F A ,! F B has a left inverse *
* r : F B ! F A,
so j* : (MBXo)intk! (MAXo)intkhas a right inverse r*, so in particular is on*
*to. Since this
is true in each simplicial dimension k, j* : MBXo ! MAXo is a fibration in G. *
*(Note that
di: Xn ! Xn-1 is always a fibration.)
(b) In addition, kn= j* : M0nXo ! MknXo is a fibration for all 0 k n, as o*
*ne can see
by considering (2.5)(since ffin-1 surjects onto the identity component by assum*
*ption).
(c) Given j 2 Cn(ssffXo), represented by h : M ! Yn, with djh ~ 0 fo*
*r 1 j n,
note that for 1 k n, MknXo is the pullback of
(dk;:::;dk)k ffikn-1
Mk+1nXo -----! Mn-1Xo --- Xn-1;
in which (dk; : :;:dk) is a fibration by (a) if k 1, so this is in fact a *
*homotopy pullback
square (see [Mat , x1]). By descending induction on 1 k n, (starting with ff*
*inn= dn), we
may assume ffik+1nO h : M ! Mk+1nXo is nullhomotopic in C, as is dkO h, *
*so the induced
pullback map, which is just ffiknO h : M ! MknXo, is also nullhomotopic b*
*y the universal
property. We conclude that ffi1nO h ~ 0, and since ffi1n: Xn ! M1nXo is a fibra*
*tion by (b), we
can replace h by a homotopic map h0: M ! Xn such that ffinh0= 0. Thus *
*h0 lifts to
ZnYo = Fib(ffin), so ? is surjective.
(d) Even though the retraction r : F [n] ! F 0nin (a) is not canonical, it ma*
*y be chosen
independently of the internal simplicial dimension k to yield a section r* for*
* ffi1n= j* : Xn!!
*
* ffi0n1
M1nXo. The long exact sequence in [M; -] for the fibration sequence CnYo i*
*-!Yn -! MnYo
(cf. [Q1 , I,x3]) then implies that i# is monic, so ? is, too. The argument *
*lifts from D = G
to C because the objects M are in the image of the adjoint of U : C ! D, by*
*_assumption
2.13. |_*
*_|
This Lemma, together with (2.27), yields a commuting diagram:
(d0)#
ssFCn+1Xo____________-ssFZnXo______-_-^ssnXop
| | ppp
| | pp
|~= ^ | ppph
? | ?| pp
| | ppp
|? dssFXo0 |? |?
Cn+1(ssFXo)__________-Zn(ssFXo)____-_-ssnssFXo
Figure 1
which defines the dotted morphism of F-algebras h : ^ssnXo! ssn(ssFXo) (this wa*
*s called the
"Hurewicz map" in [DKS2 , 7.1]). Note that for n = 0 the map ^?is an isomorp*
*hism, so h
is, too.
2.30. An exact couple. If Xo 2 sC is Reedy fibrant, the long exact sequences *
*(2.24)for
the fibrations Cn+1Xo ! ZnXo fit into an (N; ^F)-bigraded exact couple (D1*;*
*ff; E1*;ff) with
D1k;ff~=ssffZkXo and E1k;ff~=ssffCkXo for k 0 and M 2 ^F. As in [DKS2*
* , x8] the
10 DAVID BLANC
derived couple has D2k;ff~=(^sskXo)ffand E2k;ff~=ssk(ssffXo) (using Lemma 2.29)*
*, which fit into
a "spiral exact sequence"
(2.31) . .!.ssn+1ssFXo @-!^ssn-1Xos-!^ssnXoh-!ssnssFXo ! . .^.ss0Xoh-!ss0ssFX*
*o ! 0
as in [DKS2 , 8.1], so by Reedy fibrant replacement (x2.22), one has such an ex*
*act sequence for
any Yo 2 sC. Of course, ^ss-1Xo:= 0; and at the right hand end we have h : ^ss*
*0Xo~=ss0ssFXo,
as noted above.
We immediately deduce the following
2.32. Proposition.A map f : Xo ! Yo in sC is a weak equivalence in the resoluti*
*on model
category - i.e., induces an isorphism in ssnssF for all n 0 (x2.16(i)) *
*- if and only if it
is an F-equivalence - i.e., induces an isomorphism in ^ssnfor all n 0 (s*
*ee x2.22).
2.33. Resolutions. By Definition 2.18, a resolution of an object X 2 C is a sim*
*plicial object
Qo over C which is cofibrant and has a weak equivalence f : Qo ! c(X)o. Note *
*that such
an f is detemined by an augmentation " : Q0 ! X in C (with d0 O " = d1 O "*
*); by
Proposition 2.32, f is a weak equivalence if and only if the augmented F^-grade*
*d simplicial
group "* : ssFQo ! ssFX is acyclic (i.e., has vanishing homotopy groups in al*
*l dimensions
0).
The long exact sequence (2.31)then implies that
(2.34) ^ssnQo~=nssFX for alln 0:
2.35. Definition.A CW complex over a pointed category C is a simplicial object *
* Ro 2 sC,
together with a sequence of objects Rn (n = 0; 1; : :):- called a CW basis for*
* Ro - such that
Rn = Rnq LnRo (x2.4), and di|Rn= 0 for 1 i n. The morphism dn0: Rn ! Zn-1Ro *
*is
called the n-th attaching map for Ro (compare [Bl1, x5]).
A CW resolution of a simplicial F-algebra Ao is a CW complex Go 2 sF -Alg, *
* with
CW basis (Gn )1n=0such that each Gn is a free F-algebra, together with a weak *
*equivalence
OE : Go ! Ao.
2.36. Definition.In the situation of x2.7, a simplicial object Ro 2 sC is ca*
*lled a CW
resolution of Xo 2 sC if Ro is a CW complex with each Rn in F, up to homoto*
*py (so in
particular Ro is indeed cofibrant), equipped with a weak equivalence f : Ro ! X*
*o.
2.37. Remark.It is easy to see that one can inductively construct a CW resoluti*
*on for every
simplicial F-algebra Ao, since in order for OE : Go ! Ao to be a weak equiv*
*alence it is
necessary and sufficient that ZnOE take ZnGo onto a set of representatives of s*
*snAo in ZnAo,
and the attaching map dn0 map Gn onto a set of representatives for Ker(ssnOE) *
*in Zn-1Go.
Thus we can let Gn be the free F-algebra (x2.8) generated by union of the unde*
*rlying sets
of ZnAo and Ker(Zn-1f), say.
The "topological" version of this requires a little more care. In particular, *
*[Bl8, Remark 3.16]
implies that not every free simplicial F-algebra Ao is realizable in the sense*
* that there is a
Ro 2 sC with ssFRo ~=Ao. In order to see what can be said on this context, a*
*ssume given
a fibrant and cofibrant simplicial object Po with an augmentation " : P0 ! X.*
* For each
ff 2 ^F, consider the long exact sequence
(dm0)# @m-1 (jm)#
(2.38) : :s:sff+1Cm Po ---! ssff+1Zm-1Po ---!ssffZm Po ---! ssffCm Po: : :
for the fibration dm0, where Z0Po := P0. By definition, Po ! X is a resolution*
* if and only if
ssissFPo = 0 for each i 0, where the homotopy groups are understod in the augm*
*ented sense
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 11
- that is, ss0ssFPo := Ker((d00)# : C0ssFPo ! Z-1ssFPo)= Im((d10)# : C1ssFPo ! *
*Z0ssFPo). The
key technical fact we shall need in this context is contained in the following
2.39. Lemma. An fibrant and cofibrant Po 2 sC with an augmentation Po ! X *
*is a
resolution of X if and only if for each m > 0:
(jm)#
(a) There is a short exact sequence 0 ! Im(@m-1) ,! ssFZm Po ---! Zm ssFPo ! 0*
*, and
(b) @m |Im(@m-1)is one-to-one, and surjects onto Im(@m ), and Im@0 ~=ssFX.
Note that since @m shifts degrees by one, (a) and (b) together imply that I*
*m(@m ) ~=
m+1ssFX for each m.
Proof.For any Po, the inclusion jm : Zm Po ! Cm Po induces a map of F-algebr*
*as (jm )# :
ssFZm Po ! ssFCm Po ~=Cm ssFPo (see Lemma 2.29), which factors through Zm ssFP*
*o. Denote
the boundary map for the chain complex C*ssFPo (which computes ss*ssFPo) by*
* Dm :=
(jm-1)# O (dm0)#.
If Po ! X is a resolution, we must have Im((jm )# O (dm+10)#) = Im(Dm+1) = *
*Ker(Dm )
for each m 0, so in particular (jm )# maps onto Zm-1ssFPo. Moreover, since s*
*sFC1Po !
ssFP0!!ssFX ! 0 is exact, Im@0 ~=ssFX and so if we assume by induction that *
*(b) holds
for m - 1, we see that Ker(jm )# = Im@m-1 is isomorphic to m ssFX, which *
*proves (a).
Moreover, if 0 6= fl 2 Ker@m = Im(dm+10)#, and fl 2 Im @m-1 = Ker(jm )#, th*
*en we have
fi 2 ssFCm+1Po with (dm+10)#(fi) = fl 6= 0 but Dm+1(fi) = 0 - contradicting (a)*
* for m + 1.
Finally, if (jm )#(fl) 6= 0, there is a fi 2 ssFCm+1Po with Dm (fi) = (jm )#(fl*
*), by the acyclicity
of ssFPo, so fl - (dm+10)#(fi) 2 Ker(jm )# = Im @m-1, and @m (fl - (dm+10)*
*#(fi)) = @m (fl),
which proves (b) for m. The identification of Im @0 is immediate from (2.38).
Conversely, if (a) and (b) are satisfied for all m, for any element in i 2 Zm*
* ssFPo, we have
i = (jm )#(fl) for some fl 2 ssFZm Po. Thus there is a 2 ssFZm-1Po with @*
*m (@m-1()) =
@m (fl), by (b), so fl . @m-1()-1 is in Ker@m = Im(dm+10)#; thus (jm )#(fl*
* ._@m-1()-1)_= i
bounds, and ssFPo is acyclic. *
* |__|
It should be pointed out that the fundamental short exact sequence
(jm)#
(2.40) 0 ! m ssFX ~=Im(@m-1) ,! ssFZm Po ---!! Zm ssFPo ! 0
for a resolution Po is actually split, as a sequence of graded groups, because *
*(jm )#|Im(dm+10)#=
(jm )#|Ker@m is one-to-one, by (b), and surjects onto Zm ssFPo by the acyclic*
*ity. However,
Im(dm+10)# = Ker@m need not be a sub-F-algebra of ssFZm Po, since @m is not a*
* morphism
of F-algebras.
With the aid of Lemma 2.39 we can now show:
2.41. Proposition.Under the assumptions of x2.7 and 2.13, any X 2 C has a CW re*
*solution
Ro 2 sC.
Proof.Let Qo 2 sC be the functorial resolution of x2.19; we may assume that the*
* augmentation
"Q : Q0 ! X is a fibration.
We start off`by`choosing a set T*0 ssFQ0 of F-algebra generators (x2.8), such*
* that if we
let R00:= ff2F^fi2T0ffMfi, then "Q# maps the free F-algebra ssFR00 ss*
*FQ0 onto
ssFX. We may assume T*0 is minimal, in the sense that no sub-graded set gener*
*ates a free
F-algebra surjecting onto ssFX - so that "Q#(fi) 6= 0 for all fi 2 T*0.
The inclusion OE : ssFR00,! ssFQ0 defines a map f00: R00! Q0 with (f00)# = OE,*
* and we let
0 Q 0 R0 0 i "R *
* Q
"R := " O f0; factoring " by 2.1(3) as R0 -!R0 -! X and usng the LLP for*
* i and "
yields f0 : R0 ! Q0 commuting with ".
12 DAVID BLANC
Now assume by induction that we have constructed a fibrant and cofibrant Ro *
*through
simplicial dimension n - 1 0, together with a map trn-1f : trn-1Ro ! trn-1Q*
*o which
induces an embedding of F-algebras (trn-1f)#. We assume that Ro satisfies (a*
*) and (b)
of Lemma 2.39 for 0 < m < n (and of course Qo satisfies them for all m > 0)*
*. If we map
the short exact sequence (a) for Ro to the corresponding sequence for Qo by *
*f*, we see
that Zn-1(f#) = Zn-1OE : Zn-1ssFRo ! Zn-1ssFQo is one-to-one, so (Zn-1f)# : s*
*sFZn-1Ro !
ssFZn-1Qo is, too.
Any non-zero element in Zn-1ssffRo is represented by fl 2 ssFZn-1Ro, by (2.40)*
*for Rn-1.
Let g : M ! Zn-1Qo represent f#fl 2 ssffZn-1Qo, with M(g) the corr*
*esponding
coproduct summand of Qn = LQn-1 in (2.20), with i(g): M(g)! Qn the incl*
*usion.
Then diO i(g)= i(di-1g)for 1 i n (in the same notation) and d0O i(g)= g, *
* by x2.19.
Thus the F-algebra generator * 2 ssffQn is in CnssFQo, and (dn0)#** = *
*f#fl.
Thus if we choose a set T*n of F-algebra generators for Zn-1ssFRo and set
a a
(2.42) Rn := M(fi);
ff2F^fi2Tnff
we have maps fn : Rn ! CnQo and d0 : Rn ! Zn-1Ro such that (jn-1)# O (dQ0)# O (*
*fn)# =
(jn-1)# O (Zn-1f)# O (d0)#. Now (2.40)implies that (jn-1)# is one-to-one on *
* Imd0, so
(dQ0)# O (fn)# = (Zn-1f)# O (d0)#. Because (dQ0)# is a fibration and ssFRn *
* is free, this
implies that one can choose fn so that dQ0O fn= Zn-1f O d0. Since Lnf : LnR*
*o ! LnQo
exists by the induction hypothesis, one can define fn : Rn ' LnRo q Rn ! Qn *
*extending
trn-1f to trnf : trnRo ! trnQo, with ffiRn: Rn ! MnRo a fibration. Since *
*ssissFPo_= 0
then holds for i n - 1, (2.42)and (2.40)hold for m = n. *
* |__|
2.43. Remark.We have actually proved a little more: given any minimal simplicia*
*l CW res-
olution of F-algebra's Ao ! ssFX (x2.35) of a realizable F-algebra, one can *
*find a CW
resolution Ro ! X realizing it: that is, ssFRo ~=Ao. (Minimality here is under*
*stood to mean
that we allow no unnecessary F-algebra generators in each An , beyond those nee*
*ded to map
onto Zn-1Ao.)
By a more careful analysis, as in [Bl8, Thm. 3.19], one could in fact show tha*
*t any CW
resolution of ssFX is realizable. However, this will follow from Corollary 4.1*
*1 below.
3.Postnikov systems and the fundamental group action
We now describe Postnikov systems for simplicial objects in the resolution mod*
*el category,
and the fundamental group action on them.
3.1. Definition.If C is a category satisfying the assumptions of x2.7, a Postni*
*kov system for
an object Yo 2 sC is a sequence of objects PnXo 2 sC, together with maps 'n : *
*Xo ! PnXo
and pn : Pn+1Xo ! PnXo (for n 0), such that ^sskpnand ^ssk'nare isomorphi*
*sms for all
k n, and s^skPnXo= 0 for k n + 1
3.2. Remark.In general, such Postnikov towers may be constructed for fibrant X*
*o using a
variant of the standard construction for simplicial sets (cf. [May , x8]) due t*
*o Dwyer and Kan
in [DK2 , x1.2], and for arbitrary Xo by using a fibrant approximation.
Note that if Qo 2 sC is a resolution of some X 2 C (see x2.33), then by *
*(2.34)
^ssiPnQo~=issFX for n i 0, and s^siPnQo= 0 for i > n; so (2.31)implies that
8
>:
0 otherwise.
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 13
3.4. Postnikov towers for resolutions. It is actually easier to construct a cof*
*ibrant version of
the Postnikov tower for a resolution than it is to construct the resolution i*
*tself: Given a CW
resolution Qo of an object X 2 C, (constructed as in Proposition 2.41), w*
*ith CW basis
(Qk )1k=0, we construct a CW cofibrant approximation Yo !`Q(n)oas`follows.
Let J* := ssFX, and choose some G 2 hoF (i.e., G ' ff2F^ TffM) h*
*aving a
surjection of F-algebras OE : ssFG!!n+1J*. Set Yn+2:= Qn+2 q G, with (d0*
*|G)# = OE,
mapping onto n+1J* ~= Im(@n) ,! ssFZn+1Qo = ssFZn+1Yo (see (2.40)). This *
*defines
ffin+2 *
* n+2
Yn0+2:= Yn+1q Ln+2Yo --! Mn+2Yo, which we then change into a fibration. Sinc*
*e (d0 )# :
ssFCn+2Yo!!ssFZn+1Yo is surjective, we may assume by induction on k n + 2 th*
*at
~=
(3.5) (jk)* : ssFZkYo -!ZkssFYo and @k-1= 0;
and thus we may choose Yk+12 hoF with d0 : ssFYk+1!!ssFZkssFYo, and see that*
* (3.5)holds
for k + 1 by (2.40).
Note that Yo ' Q(n)ois constructed by "attaching cells" to Qo, as in the tra*
*ditional method
for "killing homotopy groups" (cf. [Gr, x17]), so we have a natural embedding*
* ae : Qo ,! Yo,
rather than a fibration. In fact, it is helpful to think of PnXo as a homotop*
*y-invariant version
of the (n + 1)-skeleton of Xo: starting with trn+1Xo, one completes it t*
*o a full simplicial
object by a functorial construction which (unlike the skeleton) depends only *
*on the homotopy
type of Xo.
3.6. -algebras and the fundamental group. Under our assumptions, the category*
* C =
F -Algis a CUGA, or category of universal graded algebras (see [BS , x2.1] a*
*nd [Mc1 , V,x6]),
so that sC, the category of simplicial F-algebras, has a model category struc*
*ture defined by
Quillen (see [Q1 , II, x4]). Equivalently, one could take the resolution mod*
*el category on sC,
starting with the trivial model category structure on F -Alg, and letting FF*
* -Algbe the sub-
category of all free F-algebras - as in x2.17. One thus has a concept of "sp*
*heres" in sF -Alg
- namely, ssFnM, for ff 2 ^F(cf. x2.22) - and (ssnAo)ff~=[nM; Ao]sF -*
*Algfor any
simplicial F-algebra Ao. Thus if we take homotopy classes of maps between (*
*coproducts
of) these spheres as the primary homotopy operations (see [W , XI, x1]), we c*
*an endow the
homotopy groups ss*Ao = (ssiAo)1i=0of Ao with an additional structure: that o*
*f a (F -Alg)-
-algebra, in the (somewhat unfortunate, in this case) terminology of [BS , x3*
*.2]. By definition,
this structure is a homotopy invariant of Ao.
In our situation, however, because we are dealing with Postnikov sections, b*
*y (3.3)we only
need the very simplest part of that structure - namely, the action of the f*
*undamental group
ss0Ao on each of the higher homotopy groups ssnAo.
Observe that because C has an underlying group structure, by assumption 2.13*
*, the indexing
of the homotopy groups of an object in sC should be shifted by one compared *
*with the usual
indexing in T*, so that ss0Ao is indeed the fundamental group, and in fac*
*t the action we
refer to is a straightforward generalization of the usual action of the funda*
*mental group of a
simplicial group (or topological space) on the higher homotopy groups.
3.7. J*-modules and J*-algebras. We shall be interested in an algebraic desc*
*ription of this
action: that is, we would like a category of universal algebras which model t*
*his action, in the
same sense that -algebras model the action of all the primary homotopy operat*
*ions on the
homotopy groups of a space. Just as in the case of ordinary -algebras, the ac*
*tion in question
is determined by the homotopy classes of maps of simplicial F-algebras.
Thus we are led to consider two distinct "varieties of algebras", in the ter*
*minology of [Mc1 ,
V, x6]): one modeled on the homotopy classes of maps, and one on the actual m*
*aps.
14 DAVID BLANC
3.8. Definition.Given a F-algebra J*, let J*-Mod denote the category of uni*
*versal
algebras whose operations are in one-to-one correspondence with homotopy classe*
*s of maps
ssFnM ! ssF(nM q 0M), and whose universal relations correspond *
*to the
relations holding among these homotopy class in ho(sC). These model ssnAo, wit*
*h the action
of ss0Ao, for Ao 2 sF -Alg.
An object K* 2 J*-Mod is itself a F-algebra, equipped with an action of an o*
*peration
: Jff00x Kff0! Kfffor each 2 [ssFnM; ssF(nM q 0M)]. Such a K**
* will be
called a J*-module, even though in general the category of such objects, which *
*we shall denote
by J*-Mod, need not be abelian (and it could depend on n). However, in the c*
*ases that
interest us, J*-Mod will be abelian, and will not depend on n > 0.
3.9. Definition.Given a F-algebra J*, let J*-Alg denote the category of univ*
*ersal al-
gebras whose operations are in one-to-one correspondence with actual maps ssFn*
*M !
ssF(nM q 0M) as above, and whose universal relations correspond to t*
*he relations
holding among these maps in sC. The objects in J*-Alg, which are again F-algeb*
*ras with
additional structure, will be called J*-algebras.
The category J*-Alg is generally very complicated; it is not abelian, and we c*
*annot expect to
know much about it, even for C = G, say. In particular, one may well have a dif*
*ferent category
for each n > 0 (although we surpress the dependence on n to avoid excessive not*
*ation). Note,
however, that maps ` : ssFnM ! Ao, for any simplicial F-algebra Ao, cor*
*respond
to elements in ZnAo, so that the A0-algebra structure on An restricts to a*
*n action of of
Z0Ao = A0 on ZnAo.
3.10. Remark.Let Qo be a resolution (in sC) of some object X 2 C, with J* := ss*
*FX, and
Yo ' PnQo its n-th Postnikov approximation. Then we have an action of ss0ssFYo*
* ~=J* on
ssn+2ssFYo ~=n+1J* which is a homotopy invariant of Yo, and thus in turn of *
*Qo, so of X.
It is not clear on the face of it whether the J*-module nJ* depends only on J*,*
* though we
shall see (in x4.5 below) this holds for n = 1, and hope to show in [BG ] that *
*in fact this holds
for all n. In any case it is describable purely in terms of the primary F-alge*
*bra-structure of
J*.
In general, for any simplicial object Xo 2 sC, there is an action of s^s0Xo~=s*
*s0ssFXo on the
higher F-algebras ^ssnXo, defined similarly via homotopy classes of maps [SnF; *
*S0FqSnF]sC (see
x2.22); but there is no reason why this should define the same category of "^ss*
*0Xo-modules" as
that defined above. Thus we do not know (2.31)to be a long exact sequence of ^s*
*s0Xo-modules.
However, in our case, when Xo = Qo is a resolution, the isomorphism of (abelian*
*) F-algebras
ssn+2ssFYo ~=n+1J* is defined inductively by means of the connecting homomorphi*
*sm of (2.31),
and this yields the J*-module structure on nJ*.
3.11. Assumption.Under mild assumptions on the category C one may show that for*
* any
Ao 2 sF -Alg and n 1, the F-algebra ssnAo is abelian (see [BS , Lemma 5.2.1]).
However, we shall need to assume more than this: namely, that J*-Mod as defin*
*ed above
is in fact an abelian category. We also assume that when Ao is a simplicial F-*
*algebra, the
action of ss0Ao on each ssnAo is induced by an action of A0 on An, and if *
* Ao = ssFQo,
then this in turn is induced by an action of Q0 on Qn. Moreover, ZnAo and *
*CnAo are
sub-A0-algebras of An, and d0 is a homomorphism of A0-algebras.
3.12. Proposition.These assumptions are satisfied for the categories listed in *
*x2.14.
Proof.As we shall see, all the categories in question are essentially special c*
*ases of the first:
(I) When C = G, the fundamental group action has an explicit description as fol*
*lows:
We define the generalised Samelson product of two elements x 2 Xp;ky 2 Xq*
*;`(where,
as in x1.2, p is the "external" dimension, k the "internal" dimension in a *
*a bisimplicial
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 15
group Xo;o2 sG) to be the element <> 2 Xp+q;k+`
0 "(oe*
*) 1
Y Y
(3.13) <> := @ (sextaeq:s:e:xtae1sint:`:s:intx1; sextoeq:s:e:*
*xtoe1sint'`:s:i:nt'1y)"(')A:
(oe;ae)2Sp;q('; )2Sk;`
Here Sp;qis the set of all (p; q)-shuffles - that is, partitions of *
*{0; 1; : :;:p + q - 1}
into disjoint sets oe1 < oe2 < . .<.oep, ae1 < ae2 < . .<.aeq - and "(oe) i*
*s the sign of the
permutation corresponding to (oe; ae) (see [Mc2 , VIII, x8]); Sp;qis ordere*
*d by the reverse
lexicographical ordering in oe. (a; b) denotes the commutator a . b . a-*
*1 . b-1 (where .
is the group operation). When p = q = 0, <> is just the usual Same*
*lson product
in X0;o2 G (cf. [C , x11.11]).
We are mainly interested here in the case p = 0, so <> := <^x;>y*
* for ^x:=
sq-1. .s.0x 2 Xq;k. It is sometimes convenient to think of this as an "act*
*ion" of x on y,
setting tx(y) := <> . y (cf. [W , X, (7.4)]).
The simplicial identities imply that if dintix = dintiy = 0 for all i, *
* the same holds for
, and if x = dint0z for some z 2 Cintk+1Xp;o, then = dint0, so that << ;>>
induces a well-defined operation << ;>> : ssintkXp;oxssint`Xq;o! ssintk+`Xp*
*+q;o, which is defined
for any simplicial -algebra Ao*, with ff 2 Ap* and fi 2 Aq*, by:
Y
(3.14) <fi> := :s:oe1fi"(oe)2 Ap+q*:
(oe;ae)2Sp;q
Again when p = 0 we write off(fi) := <fi> . fi, so that off: Aq*
**! Aq* is a group
homomorphism in each degree (if ff 2 ZpAo*, fi 2 ZqAo*, then <fi> 2 Zp*
*+qAo*).
Now let Xo;o:= 0Sk q nS`, (where Sk is the k-sphere for G - x2.17) *
*and let
0;k and n;`be -algebra-generators for sskX0;oand ss`S` ss*Xn;o, respect*
*ively, so
ss*Xn;ois generated by {^0;k; n;`}. Since djn;`= 0 (0 j n), we have a sh*
*ort exact
sequence of -algebras
(3.15) 0 ! Znss*Xo;o! ss*(Skq S`) ! ss*Sk ! 0:
When k; ` > 0, by [H , Theorem A] any element x 2 ss*Xn;o~=ss*(Sk q S`*
*) can be
written as a sum of elements of the form i# !(^0;k; n;`) (where !(x; y) = <*
*: :<:x;>y;>:i:s:
some iterated Samelson product), so x can be obtained by means of the "inte*
*rnal" -Alg
operations from expressions of the form off(n;`) (for ff 2 ss*X0;o).
By passing to universal covers we have a similar description when ` > k *
*= 0, since
then any x 2 ssjXn;o (j 1) can be written as a sum of elements of the fo*
*rm
i# !(off1(n;`); : :;:offr(n;`) (for ffi2 ss*X0;o), and any other ff 2 ss*X0*
*;oacts on this by
permuting the generators offi(n;`, so again tauff(-) is a group homomor*
*phism. When
k = ` = 0, we are reduced to the case C = Gp (see (II) below).
When k > 0 and ` = 0, let us write 'ff(fi) := <ff> for ff 2 s*
*s*X0;o and
fi 2 ss0Xn;o, so that we are thinking of the usual (internal) action of th*
*e fundamental
group ss0Xn;oas a function of fi. This is not a homomorphism, since we hav*
*e 'ff(fi .fl) =
'ff(fi) + 'ff(fl) + <^ff>>> by [W , III, (1.7) & X, (7.4)].
But <fi> is a cycle (i.e., in Znss*Xo;o), by (3.15), so <<<fi>;*
*>fl> ~ 0 in ssnss*Xo;o
for any fl 2 ss0Xn;oby [BS , 5.2.1], which means that 'ffinduces a homom*
*orphism on
ssnss*Xo;o.
In summary, an J*-algebra (x3.9), for any J* 2 -Alg, is just a -algebra *
*K* together
with an action of each ff 2 J*, which may be expressed in terms of the (deg*
*ree-shifting)
16 DAVID BLANC
homomorphisms off, or the functions 'ff, respectively, satisfying whatever*
* relations hold
among these (and the internal -algebra operations) in ss*Xn;o.
A J*-module, on the other hand, is an abelian -algebra K*, together w*
*ith homo-
morphisms off: K* ! K* or 'ff: K* ! K* for each ff 2 J*, satisfying th*
*e identities
occuring in ssnss*Xo;o.
These identities could be described more or less explicitly in the categ*
*ory -Alg, in
terms of suitable Hopf invariants (cf. [Ba1, II, x3]). Compare [Ba2, x3]).
(II) When C = Gp, sC models the homotopy theory of (connected) topological s*
*paces,
and J*-Mod, defined (as noted above) through the usual action of the fun*
*damental
group, is equivalent to the category of (left) modules over the group ring*
* Z[ss0Ao] (for
Ao 2 sC G).
(III)When C = Lie, the situation is similar to C = Gp, with Samelson products r*
*eplaced by
Lie brackets.
(IV) When C = dL sLie, one has a generalized Lie bracket defined for bisimpl*
*icial Lie
algebras as in (3.13), with commutators replaced by Lie brackets (see [Bl7*
*, x2.6]).
(V) When C = R-Mod , sC is equivalent to the category of chain complexes over*
* R, so there
is no action of ss0 = H0 on the higher groups.
__
|__|
3.16. Remark.It is possible to write down general conditions on category of un*
*iversal alge-
bras (or CUGA) C, defined in terms of operations and relations, which suffice *
*to ensure that
assumptions 3.11 hold: all one really needs is a suitable Hilton-Milnor theore*
*m in sC (see,
e.g., [Go ]). However, it seems simpler to state the conditions needed as abov*
*e, and verify them
directly in any particular case of interest.
4. Cohomology of F-algebras
In this section we complete the description of the algebraic invariants used *
*to distinguish
homotopy types. To do so, we recall Quillen's definition of cohomology in a mo*
*del category, in
the context of F -Alg:
4.1. Definition.Let C be a model category with an abelianization functor Ab :*
* C ! Cab,
where Cab denotes of course the full category of abelian objects in C; we sh*
*all usually write
Xab for Ab(X) (see x2.10). In [Q1 , II, x5] (or [Q4 , x2]), Quillen defines*
* the homology of an
object X 2 C to be the total left derived functor LAb of Ab, applied to *
*X (cf. [Q1 , I,
x4]). Likewise, given an object M 2 Cab=X, the cohomology of X with coeffici*
*ents in M is
R HomCab=X(X; M) := Hom Cab=X(LAb X; M).
4.2. Quillen cohomology of F-algebras. When J* 2 C = F -Alg, we have the mo*
*del
category structure defined in x3.6 above, so we can choose a resolution Ao ! J*
** in sF -Alg
as in x2.33, and define the i-th homology group of J* to be the i-th homotopy *
*group ssi(Ab Ao)
of the ^F-graded simplicial abelian group (Ao)ab - i.e., of the associated c*
*hain complex (cf.
[D , x1]). One must verify, of course, that this definition is independent of*
* the choice of the
resolution Ao ! J*.
Similarly, if K* is an abelian J*-algebra, then the i-th cohomology group *
*of J* with
coefficients in K*, written Hi(J*; K*), is that of the cochain complex cor*
*responding to the
cosimplicial ^F-graded abelian group Hom J*-Alg(Ao; K*).
4.3. Remark.Here Hom J*-Alg(A; B) is the group of F-algebra homomorphisms whic*
*h respect
the J*-action; because we are mapping into an abelian object K*, Hom J*-Alg*
*(Ao; K*) ~=
Hom J*-Alg((Ao)0ab; K*) (where A0abdenotes the abelianization of A 2 J*-Algas *
*an J*-algebra).
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 17
However, in the simplicial abelian J*-algebra (Ao)0abwe have a direct product *
*decomposition
(Ak)0ab= (A^0k)ab (LkAo)0abfor k 0, where (A^k)0ab:= Ck(Ao)0abis the the su*
*b-abelian J*-
algebra of (Ak)0abgenerated by (Ak)0ab(cf. [May , Cor. 22.2]) - and in fact (d^*
*0)0ab: (A^n)0ab!
(An-1)0abfactors through a map @^n: (A^0n)ab! (A^n-1)0ab(see [May , p. 95(i)]).
Thus the n-cochains split as:
Hom J*-Alg((An)0ab; K*) ~=Hom J*-Alg((A^n)0ab; K*) Hom J*-Alg(Ln(Ao)0a*
*b; K*);
so by [BK , X, x7.1] any cocycle representing a cohomology class in Hn(J*; K*)*
* may be
represented uniquely either by a map of abelian A0-algebras ^f: (A^n)0ab! K*,*
* or by a map
of A0-algebras f : An ! K*.
Since CnAo contains the sub-A0-algebra of An generated by An (by assumptio*
*n 3.11),
f determines its restriction f|CnAo: CnAo ! K*, which determines ^f, which *
*determines
f in turn. We have thus shown that H*(J*; K*) may be calculated as the cohom*
*ology of
the (abelian) cochain complex HomA0-Alg(C*Ao; K*) (even though C*Ao is not i*
*n general a
homotopy invariant of Ao, in non-abelian categories).
4.4. obstructions to existence of resolutions. Given an object X 2 C, and a (*
*suitable)
simplicial resolution Ao ! J* of the F-algebra J* := ssFX, we have seen in Se*
*ction 2 that
one can construct a resolution Qo of X (in the resolution model category sC) re*
*alizing Ao,
in the sense that ssFQo ~=Ao. It is thus natural to ask whether any simplicia*
*l F-algebra -
or at least, any resolution Ao of an abstract F-algebra J* - is realizable in s*
*C.
One approach to this question in the topological setting (i.e., for C = G), in*
* terms of higher
homotopy operations, was given in [Bl3]. However, a glance at the proof of Prop*
*osition 2.41
shows that one can instead consider obstructions to extending trnQo to the nex*
*t simplicial
dimension. For a homotopy-invariant description, we state this in terms of succ*
*essive Postnikov
approximations to Qo, since it is clear that, once we have constructed trnQo*
*, it is always
possible to obtain Yo ' Q(n-1)ofrom it by successive choices of free objects *
*Yk+12 hoF
(k = n; : :):mapping to ZkYo by a F-algebra surjection.
4.5. constructing the obstruction. Assume given a CW resolution Ao 2 sF -Alg *
*of J*,
with CW basis (An )1n=0, and choose corresponding free objects Qn 2 F C with s*
*sFQn ~=An .
We begin the induction with tr1Qo, and thus Q(0)o, constructed as in the proof *
*of Proposition
2.41. Note that to obtain tr1Qo we do not in fact need to know X 2 C with s*
*sFX ~=J* -
or even to know that such an object exists! This implies that the J*-module str*
*ucture on J*
is uniquely determined.
In the inductive stage we assume given trnQo (equivalently: Q(n-1)o), satis*
*fying 2.39(a)
and (b) for 0 < m n. Our strategy is to try to attach (n + 1)-dimensional "*
*cells" to
trnQo in such a way as to guarantee acyclicity of the resulting trn+1Qo in one *
*more simplicial
dimension - using Lemma 2.39 above. The key to the construction of trn+1Qo from*
* trnQo
thus lies in the extension of A0-algebras (2.40)(for Qo, rather than Po), *
* in which the
two ends are given to us. Observe that this extension determines the A0-algebra*
* structure on
nJ*, if more than one is possible.
We want this extension to be "trivial" (that is, split as a semi-direct produc*
*t of A0-algebras),
in order to be able to lift the given map of A0-algebras dA0: An+1 ! ZnAo to*
* a map
dQ0: Qn+1 ! ZnQo, so the question is reduced from one about simplicial objects*
* over C to
one of algebraic objects, namely: A0-algebras. There is a close analagy to the*
* classical theory
of group extensions, where the triviality of an extension E : 0 ! A ! B ! G *
*is measured
by the characteristic class (E) 2 H2(G; A) (compare [Mc2 , IV, x6]). Similar*
*ly, in our case
the triviality of the extension is measured by the vanishing of a suitable coho*
*mology class in
Hn+2(J*; nJ*), defined as follows:
18 DAVID BLANC
Because (jn)# : ssFZnQo!!ZnssFQo ~=ZnAo is surjective, and An+1 is a free F-*
*algebra,
we can choose a lifting in the following diagram:
dAn+1
An+1 _________-0ZnAo___-0
ppp
ppp |~
ppp ||=
pp Q |?
i |? (jn)#
0____-nJ*____-ssFZnQo_____-ZnssFQo____-0
Figure 2
and we can find a map ` : Qn+1! ZnQo realizing (again, because An+1 = ssFQn+1 *
* is free).
Combined with the "tautological map" Ln+1Qo ! Mn+1Qo (see x2.4), which depends *
*only on
trnQo, by setting Qn+1:= Qn+1qLn+1Qo we obtain an extension d0 : Qn+1! Qn of ` *
*(which
is a map of Q0-algebras), and thus an (n+1)-truncated simplicial object trn+1Qo*
* over C, with
Qn+1:= Qn+1q Ln+1Qo, and ssF trn+1Qo ~=trn+1Ao. In particular, dQn+10: Cn+1Qo *
*! ZnQo
induces a map ^ from ssFCn+1Qo = Cn+1Ao to ssFZnQo extending (and determined b*
*y) the
lifting : An+1! ssFZnQo of dAn+10. This is a map of A0 = ssFQ0-algebras, by A*
*ssumption
3.11.
Since (jQn)# O (^|Zn+1Ao) = 0, the map ^|Zn+1Aofactors through : Zn+1Ao ! Ker*
*(jQn)# =
nJ*, and composing with dAn+20: Cn+2Ao ! Zn+1Ao defines : Cn+2Ao ! nJ* - agai*
*n,
a map of A0-algebras:
dAn+20 j dAn+10
Cn+2Ao________-Zn+1Ao_____-Cn+1Ao______-ZnAo___-0
| | |
@ | | |
@ | | |
@ | | |
| |^ |~=
@ | | |
@ | | |
@ | | |
| | |
@@R|? i |? (jQn)# |?
0______-nJ* ____-ssFZnQo______-ZnssFQo__-0
The cochain = O dAn+20is clearly a cocycle in the cochain complex Hom J*-Mod*
*(Ao; J*),
so it represents a cohomology class On 2 Hn+2(J*; nJ*), called the characterist*
*ic class of the
extension.
4.6. Lemma. The cohomology class On is independent of the choice of lifting .
Proof.Assume that we want to replace in x4.5 by a different lifting 0: An+1 !*
* ssFZnQo,
and choose maps `; `0: Qn+1! ZnQo realizing , 0 respectively; their extensions *
*to maps
Qn+1! Qn (which we may denote by d0, d00) agree on Ln+1Qo. We correspondingly h*
*aving
0: Zn+1Ao ! nJ* and 0:= 0O dAn+20. `
Because Qn+1 := Qn+1 q Ln+1Qo is a coproduct of the form iM, by x2*
*.13 the
underlying group structure on any X 2 C induces a group structure on
(4.7) Hom C(Qn+1; X)
(and similarly for Hom F -Alg(An+1; ssFX)).
Therefore, we can set h := (d0)-1. (d00) : Qn+1! Qn, and h induces a map j *
*: Cn+1Ao !
ssFZnQo such that j|An+1= -1. 0. Moreover, because d0 and d00agree outside *
*of Qn+1,
(jQn)# O j = 0. Thus j factors through i : Cn+1Ao ! nJ*, which is a map of*
* A0-algebras
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 19
because nJ* is an abelian A0-algebra (actually, a J*-module), and i is induc*
*ed by group
operations from the A0-algebra maps d0 and d00.
Moreover, i|Zn+1Ao= - 0 in the abelian group structure on Hom J*-Mod(-; nJ*) *
*(which
*
* __
corresponds to the group structure of (4.7)). Thus 0- = ^jO dAn+20is a cobound*
*ary. |__|
4.8. Theorem. On = 0 if and only if one can extend Q(n-1)oto an n-th Postnik*
*ov approxi-
mation Q(n)oof a resolution of X.
Proof.First assume that there exists Yo ' Q(n+1)owith trnYo ~=trnQo: by Lemm*
*a 2.39 we
know (jQn)#|Im(dn+10)#is one-to-one (and onto ZnssFQo), for dn+10: Cn+1Yo ! ZnY*
*o = ZnQo,
and thus Im (dn+10)# \ Im@Qn-1= {0}. But then we can choose : An+1! ssFZnQo t*
*o factor
through Im(dn+10)#, (and this will induce a map of A0-algebras because of x3*
*.10), so that
= 0 and thus = 0.
Conversely, if On = 0, we can represent it by a coboundary = # O dAn+20for*
* some A0-
algebra map # : Cn+1Ao ! nJ*, and thus get i O #|An+1: An+1! ssFZnQo. If we*
* set 0:=
. (i O #|An+1)-1, we have Im 0\ nJ* = {0}. We can therefore choose dQn+10: Qn*
*+1! ZnQo
realizing 0, and then (dQn+10)# avoids Im (@Qn-1) ~=nJ*, so that trn+1Qo so co*
*nstructed
yields Q(n+1)o, as required. In particular, this determines a choice of J*-mod*
*ule structure_on
n+2J* (if more than one is possible), via (2.40)for n + 1. *
* |__|
4.9. notation. If we wish to emphasize the dependence on the choice of , we sh*
*all write
Q(n+1)o[] for the extension of Q(n)oso constructed.
4.10. Proposition.The class On depends only on the homotopy type of Q(n-1)oin*
* sC.
Proof.Assume Q(n-1)ohas been constructed, realizing a simplicial resolution of*
* F-algebras
Ao ! J* through simplicial dimension n, and let Bo ! J* be any other F-alge*
*bra
resolution: we then have a weak equivalence ' : Bo ! Ao in sF -Alg. Assume by*
* induction
on 0 m < n that we have constructed an m-truncated simplicial object trmRo ove*
*r C, and
a map f : trmRo ! trmQ(n-1)orealizing trm'. Moreover, assume that we have a ma*
*p of the
(split) short exact sequences (2.40)(in dimension m) for Ro and Qo:
i (jmR)#
0____-m J*___-ssFZm Ro________-Zm ssFRo__-0
|= |(Z f) |Z (f ) = Z '
| | n # | n # n
| | Q |
|? i |? (jm )# |?
0____-m J*___-ssFZm Qo________-Zm ssFQo__-0
Now, in order to extend f to dimension n + 1, we must choose the map (dRm+1*
*0)# :
ssFRm+1 ! ssFZm Ro (lifting dBm+10: Bm+1 ! Zm Bo) in such a way that (Zm f)# O(*
*dRm+10)# =
(dRm+10)# O Zm '. Since Bm+1 = ssFRm+1 is free, it suffices to show that the *
*obvious map from
(jQm)# Zm'
ssFZm Ro to the pullback of ssFZm Qo ---! Zm ssFQo = Zm Ao --- Zm Bo is a sur*
*jection:
given (a; b) 2 ssFZm QoxZm Bo with (jQm)#(a) = '(b), for any z 2 ssFZm Ro with *
*(jRm)#(z) = b
we have an ! 2 m+1J* ssFZm Ro such that (Zm f)#(z . !) = (Zm f)#(z0) . ! = b*
* in the
diagram above (where . is the group operation), so z . ! maps to (a; b). *
*Thus we can
choose dRm+10: Rm+1 ! Zm Ro in such a way that we can define trm+1Ro, togeth*
*er with a
map trm+1f : trm+1Ro ! trm+1Qo realizing trm+1'.
Because ' was a weak equivalence of resolutions, it is actually a homotopy equ*
*ivalence, with
homotopy inverse : Ao ! Bo, say, and the above argument also yields a homot*
*opy inverse
20 DAVID BLANC
for f(m) (or trm+1f). Moreover, the characteristic classes we defined are cl*
*early functorial
with respect to maps in sC; since the characteristic class Om+1 2 Hm+3(J*; m+1J*
**), defined
for the resolution Ao ! J* by means of the lift dQm+10, must vanish, by Theo*
*rem 4.8, the
same holds for Ro, so by Theorem 4.8 again we can extend R(m)oto R(m+1)o, *
*and_continue
the induction as long as m < n. *
* |__|
We deduce the following generalization of Proposition 2.41:
4.11. Corollary.Given X 2 C, any CW F-algebra resolution Ao ! ssFX is reali*
*zable as
a resolution Qo ! X in sC.
One could further extend Proposition 4.10 to obtain a statement about the natu*
*rality of the
characteristic classes with respect to morphisms of F-algebras : J* ! L*. H*
*owever, such
a statement would be somewhat convoluted, in our setting, and it seems better t*
*o defer it to a
more general discussion of the realization of simplicial F-algebras, in [BG ].
4.12. realization of -algebras. If G : S* ! G denotes Kan's simplicial loop f*
*unctor (cf.
[May , Def. 26.3]), with adjoint W : G ! S* the Eilenberg-Mac Lane classifying*
* space functor
(cf. [May , x21]), and S : T* ! S* is the singular set functor, with adjoint *
* k - k : S* ! T*
the geometric realization functor (see [May , x1,14]), then functors
S G
(4.13) T* k-k S* G
W
induce isomorphisms of the corresponding homotopy categories (see [Q1 , I, x5])*
*, so any homo-
topy-theoretic question about topological spaces may be translated to one in G.*
* In particular,
in order to find a topological space X having a specified homotopy -algebra J**
* ~=ss*X, it
suffices to find the corresponding simplicial group X 2 G (with the F-algebra*
* J* suitably
re-indexed). If J* is realizable by such an X, any free simplicial resolution Q*
*o ! X evidently
provides a -algebra resolution ss*Qo of J* = ss*X. But the converse is also tr*
*ue: if Qo 2 sG
realizes some (abstract) -algebra resolution Ao 2 s -Alg of J*, then the co*
*llapse of the
Quillen spectral sequence of [Q2 ], with
(4.14) E2s;t= sss(sstQo) ) sss+tdiagQo
converging to the diagonal diagQo 2 G (defined (diagQo)k = (Qk)intk) implie*
*s that
ss*diagQo ~=J*. Thus J* is realizable by a simplicial group (or topological sp*
*ace) if and only
if some -algebra resolution Ao ! J* is realizable.
The characteristic classes (On)1n=0(whose existence was promised in [DKS2 , x1*
*.3] under the
name of the "k-invariants for J*), thus provide a more succinct (if less explic*
*it) version of the
theory described in [Bl3, x5-6] (as simplified in [Bl6, x6]), for determining t*
*he realizablity of a
-algebra in terms of higher homotopy operations - which we summarize in
4.15. Theorem. Given an (abstract) -algebra J*, the following conditions are *
*equivalent:
(1) J* is realizable as ss*X for some topological space X 2 T*.
(2) Any CW -algebra resolution Ao ! J* is realizable by a simplicial space Qo.
(3) The (inductively defined) characteristic classes On 2 Hn+2(J*; nJ*) (n = *
*0; 1; : :): all
vanish.
Of course, the characteristic class On+1 is determined by the choice of some e*
*xtension Q(n)o
of Q(n-1)o, so as usual our obstruction theory requires back-tracking if at s*
*ome stage we find
On 6= 0. We shall now show how we can use other cohomology classes to determin*
*e the choices
of extensions at each stage:
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 21
4.16. distinguishing between different resolutions. A more interesting question*
*, perhaps,
is how one can distinguish between non-equivalent realizations Qo; Ro 2 sC of*
* a fixed F-
algebra resolution Ao ! J* of a realizable F-algebra J* ~=ssFX. Of course, i*
*f Qo and
Ro are both resolutions (in the resolution model category sC) of weakly equiv*
*alent objects
X ' Y in the model category C, then by definition Qo is weakly equivalent *
*(actually:
homotopy equivalent) to Ro. Thus we are looking for a way to distinguish betw*
*een objects
in C, using the iterative construction of a resolution Qo ! X (or equivalentl*
*y, the Postnikov
system for Qo).
There are a number of possible approaches to this question: one could try to c*
*onstruct a
homotopy equivalence Qo ! Ro by induction on the Postnikov tower for Ro, us*
*ing an
adaptation to sC of the classical obstruction theory for spaces (cf. [W , V, x*
*5]). Alternatively,
one could try directly to construct a map Qo ! Y realizing the augmentation *
*ssFAo ! J*
(see [Bl3, x7], and compare [B , x5]). A description more in this spirit will b*
*e given in [BG ].
Here our strategy is similar to that of x4.4: rather than assuming that we are*
* given X and
Y to begin with, we try to construct all different realizations (up to homotop*
*y equivalence in
sC) of a given simplicial F-algebra Ao (which is assumed to be a resolution o*
*f a realizable
F-algebra J*). We start our construction as in x4.5, and in the induction step *
*we have assume
given trnQo - or equivalently Q(n-1)o, satisfying the assumptions of x4.5 (*
*see the proof of
Proposition 2.41). We ask in how many different ways we can attach (n + 1)-di*
*mensional
"cells" to extend the realization one further dimension.
Again the key lies in the extension of F-algebras of (2.40). Of course, we may*
* assume that
the characteristic class On 2 Hn+2(J*; nJ*) vanishes, so that it is possible to*
* find "splittings"
for (2.40), given by various liftings in Figure 2 - all of which yield the s*
*ame cohomology
class On by Lemma 4.6. As in the classical case of groups, we find that the dif*
*ference between
two such "semi-direct products" is represented by suitable cohomology classes, *
*in dimension
lower by one than the characteristic classes (see [Mc2 , IV, x2]).
4.17. Definition.Assume given two liftings ; 0: An+1! ssFZnQo in Figure 2 above*
*, which
define extensions of trnQo - so that, as in the proof of Theorem 4.8, we may as*
*sume without
loss of generality that the corresponding maps ; 0: Zn+1Ao ! nJ* vanish. As in *
*the proof of
Lemma 4.6, we extend , 0 to face maps d0; d00: Qn+1! Qn, define j : Cn+1Ao ! ss*
*FZnQo
with (jQn)# O j = 0, and lift to a map of A0-algebras i : Cn+1Ao ! nJ*. A*
*gain
i|Zn+1Ao= - 0, which is zero, so i is a cocycle in Hom J*-Mod(Ao; J*), rep*
*resenting a
cohomology class ffi;0 2 Hn+1(J*; nJ*), which we call the difference obstru*
*ction for the
corresponding Postnikov sections Q(n)o[] and Q(n)o[0] (in the notation of x4.9).
Just as in the proof of Proposition 4.10, one can show that the classes ffin+1*
*;0n+1in question
do not in fact depend on the choice of F-algebra resolution Ao ! J*, but only*
* on the
homotopy type of Q(n-1)oin sC. Their significance is indicated by the following
4.18. Theorem. If ffi;0 = 0 then the corresponding Postnikov sections Q(n)o[] a*
*nd Q(n)o[0]
are weakly equivalent.
Proof.If i is a coboundary, there is a map # : CnAo ! nJ* such that i = # *
*O dAn0.
Composing with the inclusion i : nJ* ,! ssFZnQo yields a morphism of A0-algeb*
*ras ' :
An ! ssFZnQo. If, as in the proof of Proposition 2.41, we set Q0n:= Qn q LnQo*
*, we may
realize ' by a map z0 : Q0n! ZnQo. Since we assumed Q0n is actually a copro*
*duct of
objects in ^F, it is a cogroup object in C by x2.7(i), so using the resulting*
* group structure on
Hom C(Q0n; Qn) we may set s0:= k . z : Q0n! Qn, where k : Q0n,! Qn is the inclu*
*sion. Since
k is a trivial cofbration and Qn is fibrant in C, we have a retraction r : Qn*
* ! Q0n(which is
a weak equivalence). Let s := s0O r : Qn ! Qn.
22 DAVID BLANC
Recall from x2.13 that we have a faithful forgetful functor U^: C ! D, where f*
*or simplicity
we may assume D = G or D = sR-Mod (the other cases are trivial). We therefo*
*re have a
further forgetful functor U0 : D ! S, and we denote U0O ^U simply by U : C !*
* S. The
group operation map, while not a morphism in C or D, is a map m : UQnx UQn ! UQ*
*n in
S. Thus the following diagram commutes in S:
id > U(z0O r O j) m|ZnQo
UZnQo ___________________-UZnQox UZnQo __________-UZnQo
| | |
Uj | |U(j x j) |Uj
| | |
|?Ur Uk > U(j O z0) |? m |?
UQn ____-'UQ0n____________-UQn x UQn ____________-UQn
@@__________________________________________
Us
Since U is faithful, this implies that s O j : ZnQo ! Qn factors through a m*
*ap t :
ZnQo ! ZnQo in C. Moreover, because we assumed that each M 2 ^F is of the*
* form
M = F M0 for some M02 S (where F = ^FOF 0is adjoint to U : C ! S), *
*any map
b : M ! ZnQo corresponds under the adjunction isomorphism to ^b: M0! U*
*ZnQo,
and thus t#fi = fi . (i O (jQn)#fi) for any fi 2 ssFZSnQo (since the group o*
*peration . in
ssFZnQo is induced by m - cf. [Gr, Prop. 9.9]).
Now if ` : Qn+1! ZnQo realizes , we have (t O `)# = (` . (z0O `))# = . (# O (*
*jQn)# O ) =
. (-1. 0) = 0: An+1! ssFZnQo. Thus we have a comutative diagram
0 (jQn)#
ssFQn+1 = An+1_______-ssFZnQo______-ZnssFQo = ZnAo
| | |
id| |t# |id
| | |
|? |? (jQn)# |?
ssFQn+1 = An+1_______-ssFZnQo______-ZnssFQo = ZnAo
which yields a map of (n+1)-truncated objects ae : trn+1Qo[] ! trn+1Qo[0] (or e*
*quivalently,
Q(n)o[] ! Q(n)o[0]). Clearly ae induces an isomorphism in sskssF for k n + 1.
Now for any choice of lifting we have ssn+2ssFQ(n)o[] ~=Im(@Qn), and since
(# O (jQn)#)|Im(@Qn-1)= 0;
we find (t#)|Im(@Qn-1)= id, so by 2.39(b) the diagram
@Qn
ssff+1ZnQo_______-ssffZn+1Qo
t | |id
#| |
| Q |
|? ________-@n |?
ssff+1ZnQo ssffZn+1Qo
commutes. Thus ae induces an isomorphism on Im (@Qn), so that (ae)* : Q(n)o[] !*
*_Q(n)o[0]_is a
weak equivalence. |*
*__|
4.19. Remark.Given a (realizable) F-algebra J*, a CW resolution Ao 2 sF -Al*
*g of J*,
and a fixed (but arbitrary) choice object X 2 C with ssFX ~=J*, by Corollary 4.*
*11 we have a
corresponding resolution Qo ! X. If X02 C is another realization of J* with co*
*rresponding
ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES 23
Q0o! X0, we may assume without loss of generality that Yo0:= (Q0o)(n)' Yo := Q(*
*n)ofor some
n 0, with ; 0: An+2! ssFZn+1Qo ~=ssFZn+1Q0othe respective liftings.
4.20. different realizations of a -algebra. Assume given an abstract -algebra J*
**, which
is known to be realizable (e.g., by the cohomological criterion of Theorem 4.15*
*). We wish
to distinguish between the various non-weakly equivalent realizations of J* b*
*y topological
spaces (or simplicial groups). The spectral sequence (4.14)implies that in orde*
*r for two such
X; X02 G (with ss*X ~=J* ~=ss*X0) to be weakly equivalent, it suffices that the*
*ir corresponding
resolutions Qo ! X and Q0o! X0 be weakly equivalent (and thus homotopy equiv*
*alent)
in the resolution model category. This is in fact the main reason for consideri*
*ng this model
category structure on sG in the first case (and justifies its original name o*
*f "E2-model
category" in [DKS1 ]).
Note, however, that this is not a necessary condition; an alternative model st*
*ructure on sS
(or sG), defined in [Mo ], has as weak equivalences precisely those maps in *
*sC inducing an
equivalence on the realizations.
The difference obstructions ffi;0 , which yield an inductive procedure for d*
*istinguishing
between various realizations of a given -algebra resolution Ao ! J*, thus aga*
*in provide an
alternative to the theory described in [Bl3, x7] (as simplified in [Bl4, x4.9])*
* for distinguishing
between different realizations of a given -algebra, in terms of higher homotopy*
* operations.
To state this explicitly, assume given an (abstract) -algebra J*, a CW resol*
*ution Ao 2
s -Alg of J*, and two realizations Qo; Q0o2 sG of Ao, determined as in x*
*4.16 by
successive choices of lifts k+1: Ak+1 ! ssFZkQo and 0k+1: Ak+1 ! ssFZkQ0o. B*
*y x4.12,
we know that the realizations X := diagQo and X0 := diagQ0o are two realizat*
*ions of
J*. If ffi0;00= 0, there is a weak equivalence f0 : (Q0o)(0)' Q(0)o, which*
* we can use to
push forward 01: A2 ! ssFZ1Q0o to 001: A2 ! ssFZ1Qo so it is meaningful to c*
*onsider
ffi1;01:= ffi1;0012 H2(J*; J*). Proceeding in this way we obtain the following
4.21. Theorem. Assume given a -algebra J*, a CW resolution Ao 2 s -Alg of *
*J*,
and two topological spaces X; X0 2 T* realizing J*, corresponding to X; X0*
* 2 G under
(4.13). Let Qo; Q0o2 sG be CW resolutions of X; X0 respectively, determine*
*d as in x4.16
by successive choices of lifts n+1 : An+1 ! ssFZnQo and 0n+1: An+1 ! ssFZnQ0*
*o. If the
difference obstructions ffin+1;0n+12 Hn+2(J*; n+1J*) vanish for all n 0, t*
*hen X and X0
are weakly equivalent.
Again, these classes satsify certain naturality conditions, which are more eas*
*ily stated for
simplicial F-algebras: see [BG ].
4.22. Remark.Theorem 4.21 provides a collection of algebraic invariants - sta*
*rting with the
homotopy -algebra ss*X - for distinguishing between (weak) homotopy types of sp*
*aces. As
with the ordinary Postnikov systems and their k-invariants, these are not actua*
*lly invariant,
in the sense that distinct values (i.e., non-vanishing difference obstructions)*
* do not guarantee
distinct homotopy types. Thus we are still far from a full algebraization of ho*
*motopy theory
- even if we disregard the fact that -algebras, not too mention their cohomolog*
*y groups, are
rather mysterious objects, and no non-trivial naturally occurring examples are *
*fully known to
date.
Note, however, that we have a considerable simplification of the theory in the*
* case of the
rational homotopy type of simply-connected spaces: in this case the F-algebras *
*in question
are just connected graded Lie algebras over Q, and the cohomology theory reduce*
*s to the usual
Cartan-Eilenberg cohomology of Lie algebras. The obstruction theory we define a*
*ppears to be
the Lie algebra version of the theory for graded algebras due to Halperin and S*
*tasheff in [HS ].
See also [O , xIII] and [F].
24 DAVID BLANC
Another such simplification occurs when we consider only the stable homotopy t*
*ype: in this
case F-algebras are just graded modules over the stable homotopy ring ss := ss*
*S*S0, and
the cohomology groups in question are Ext*ss(J*; -nJ*). Here we have no acti*
*on of the
fundamental group to worry about.
Furthermore, the spectral sequence of (4.14)implies that if Q(n)o~=(Q0o)(n), *
* then also
(diagQo)(n)~=(diagQ0o)(n), so one can also use the theory described above "with*
*in a range".
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Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel
E-mail address: blanc@math.haifa.ac.il
*