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THE STABLE FREE RANK OF SYMMETRY OF PRODUCTS OF SPHERES

BERNHARD HANKE

ABSTRACT. A well known conjecture in the theory of transformation groups states that ifp is aprime and(Z/p)r acts freely on a product ofk spheres, thenr ≤ k. We prove this assertion ifp islarge compared to the dimension of the product of spheres. The argument builds on tame homotopytheory for non-simply connected spaces.

1. INTRODUCTION

Transformation group theory investigates symmetries of topological spaces. An important as-pect of this program is to define and study invariants that distinguish spaces admitting lots ofsymmetries from less symmetric ones.

In this paper we concentrate on one of these invariants, the so calledfreep-rank

max{r | (Z/p)r acts freely on X} ,defined for any topological spaceX and any prime numberp . Here all groups are acting topolog-ically. We recall the following fact from classical Smith theory.

Theorem 1.1.The freep-rank ofSn is equal to

1 for odd n and all p

1 for even n and p = 2

0 for even n and p > 2 .

In view of this theorem it is natural to look for a corresponding result, ifX is not just a singlesphere, but a product of spheres,

X = Sn1 × Sn2 × . . .× Snk .The following statement appears in several places in the literature either as a question [2, Question7.2], [23, Problem 809] or as a conjecture [1, Conjecture 2.1], [3, Conjecture 3.1.4].

Conjecture 1.2. If (Z/p)r acts freely onX, thenr ≤ k.

Actually, if p is odd, it is (in view of Theorem 1.1) reasonable to conjecture thatr is boundedabove by the numberko of odd dimensional spheres inX. Conjecture 1.2, in this sharper form foroddp, has been verified in the following cases:

1) k ≤ 2, see Heller [18];k ≤ 3, p = 2, see Carlsson [10].2) n1 = . . . = nk and in addition

a) the induced action on integral homology is trivial, see Carlsson [9], orb) the induced action on integral homology is unrestricted, but ifp = 2, thenni 6= 3, 7, see

Adem-Browder [2] (forp 6= 2 or ni 6= 1, 3, 7), and Yalcın [32] (forp = 2 andni = 1).3) assumen1 ≤ n2 ≤ . . . ≤ nk. Thenn1 ≥ 2 and for all1 ≤ i ≤ k − 1 eitherni = ni+1 or

2ni ≤ ni+1. Furthermore, the induced(Z/p)r-action onπ∗(X) is trivial andp > 3 dimX. SeeSorensen [28].

Date: April 14, 2009; MSC 2000: primary 57S17; secondary 55P62.1

2 BERNHARD HANKE

The following theorem is our main result. It settles a stable form of Conjecture 1.2.

Theorem 1.3. If p > 3 dimX and(Z/p)r acts freely onX, thenr ≤ ko .

Because(Z/p)ko acts freely on a product ofko odd dimensional spheres, this implies that thefreep-rank ofX is equal toko assumed thatp > 3 dimX. Theorem 1.3 also leads to the followingestimate of thefree toral rankof X, which follows from [17, Theorem T].

Theorem 1.4. If (S1)r acts freely onX, thenr ≤ ko .

However, even for very large primes Theorem 1.3 cannot be deduced from this result by somesort of limiting process: Browder [8] constructed free actions of(Z/p)r on (Sm)k for each oddm ≥ 3 andr ≥ 4, k ≥ r, p > km/2 which are exotic in the sense that they cannot be extended to(S1)r-actions.

Theorem 1.4 is implied by a theorem of Halperin [17] which yields upper bounds of free toralranks in terms of homotopy Euler characteristics. The proof is based on rational homotopy theoryapplied to the Borel spaceX(S1)r . Hence rational homotopy theory has become an important toolin the study of torus actions in general, see [4, Section 2] for a survey.

Because the rational homotopy type ofB(Z/p)r is trivial, it is obvious that in the context ofTheorem 1.3, rational homotopy theory cannot be applied in a reasonable way to study the BorelspaceX(Z/p)r . But tame homotopytheory seems a more promising approach. This theory wasinvented by Dwyer [14] and is modelled on Quillen’s rational homotopy theory [25], but withoutimmediately losingp-torsion information for all primesp. One of the first observations in tamehomotopy theory may be phrased as follows: ifX is an (r − 1)-connected space (r ≥ 1) andπr+k(X) is aZ[p−1 | 2p − 3 ≤ k]-module for everyk ≥ 1, then the complexity of the Postnikovinvariants ofX should be comparable to that of a rational space due to the vanishing of the relevanthigher reduced Steenrod power operations in the Postnikov pieces ofX.

Nevertheless, in the original setup, Dwyer’s theory could only be formulated forr > 2, see[14, 1.5]. In the tame setting, the restriction to simply connected spaces, which excludes theBorel spaces for(Z/p)r-actions, seems unavoidable due to the non-vanishing of thep-fold MasseyproductsH1(BZ/p;Fp) → H2(BZ/p;Fp) for any odd primep, see [19, Theorem 14]. We willmake more comments on this point later on.

To any spaceX one can associate theSullivan-de Rham algebra[6, 31], a commutative gradeddifferential algebra overQ modelled on de Rham differential forms, which calculates the rationalcohomology ofX. This construction widens the scope of rational homotopy theory from sim-ply connected to nilpotent spaces. A distinctive feature of this approach is the construction of aminimal modelfor any spaceX out of the Sullivan-de Rham algebra. The minimal model still cal-culates the rational cohomology of the underlying space, but in addition its associated vector spaceof indecomposables can be identified with the dual ofπ∗(X) ⊗ Q, if X is nilpotent,H∗(X;Q) isfinite dimensional in each degree andπ1(X) is abelian.

For a proof of Theorem 1.4 one writes down a minimal model of the Borel spaceX(S1)r andargues that ifr > ko, then the cohomology of the minimal model is nonzero in arbitrarily highdegrees. But this contradicts the fact that by the freeness of the action,X(S1)r is homotopy equiva-lent to the finite dimensional spaceX/(S1)r. The analysis of the minimal model that leads to theseconclusions is carried out in the fundamental paper [17].

The generalization of the Sullivan-de Rham algebra to the tame setting is realized in the workof Cenkl and Porter [12] and is achieved by considering commutative graded differential algebrasoverQ which are equipped withfiltrations as a new structural ingredient. By definition, elementsin filtration level q are divisible by any primep ≤ q, but not necessarily by larger primes, and

THE STABLE FREE RANK OF SYMMETRY OF PRODUCTS OF SPHERES 3

filtration levels are additive under multiplication of elements. The construction of the Cenkl-Portercomplex is based on differential forms similar to the Sullivan-de Rham algebra, but in order tokeep sufficient control over filtration levels one works with differential forms defined on a cubi-cal decomposition of the standard simplex. TheCenkl-Porter theoremstates that the integrationof forms yields a cochain map of the Cenkl-Porter complex in filtration levelq to the singularcochain complex withZ[p−1 | p ≤ q]-coefficients which in cohomology induces a multiplicativeisomorphism. We will recall the essential steps of this construction in Section 3 of our paper. Ourproof of the Cenkl-Porter theorem is different from the original approach in [12] in that we do notanalyse the integration map, which is not multiplicative on the cochain level, but construct multi-plicative cochain maps inducing isomorphisms in cohomology from the Cenkl-Porter and singularcochain complexes to a third cochain complex. This approach is inspired by the discussion ofthe de Rham theorem in [16, Theorem II.10.9] and enables us to take up and resolve the issue ofpossibly non-vanishing higher Massey products at the end of Section 3.

It is now desirable to replace the Cenkl-Porter complex by a smaller filtered commutative gradeddifferential algebra which nicely reflects the homotopy type of the underlying space in a similarmanner as the minimal algebra does in rational homotopy theory. In the tame setting, the mosttransparent and elementary approach to the construction of such an algebra is based on atameHirsch lemmawhich is used to build the desired small filtered cochain algebra along a Postnikovdecomposition of the underlying space, compare the discussion in [13] for the rational case. Forsimply connected spaces such a result is proven in [26, p. 203]. The non-simply connected case,which is relevant for our purposes, is much more involved and was carried out in the remarkablediploma thesis of Sorensen [28]. After introducing the necessary notions we will state the tameHirsch lemma in Section 4. The proof of this result, which is not available in the published liter-ature, will be given in an appendix to our paper along the lines of [28]. I owe my thanks to TillSorensen, who left academia some seventeen years ago, for producing a PDF scan of his work andto the FU Berlin for storing it on their preprint server.

With this machinery in hand we construct small approximative commutativeFp-cochain modelsof Borel spaces associated to(Z/p)r-spacesX under fairly general assumptions in Section 5. Themain idea is to use a simultaneous Postnikov decomposition of the fibre and total space of theBorel fibrationX ↪→ X(Z/p)r → B(Z/p)r.

The cochain model resulting from this discussion is the starting point for our proof of Theorem1.3 in Section 2. The argument is inspired in part by the paper [17], which provides techniques tosimplify the analysis of free commutative graded differential algebras over fields of characteristiczero. We emphasize that as stated, some of the arguments inloc. cit. do not work over fields ofprime characteristics. However, in the context of actions on products of spheres, with which weare mainly concerned, the resultingFp-cochain algebras are special enough so that the proof ofTheorem 1.3 can be completed fairly quickly by a direct argument.

The main result of Section 5, Theorem 5.6, may be useful for other purposes in the cohomologytheory ofp-torus actions, which is now closely tied to the cohomology theory of(S1)r-actions, ifp is large. For example we hope that it is possible to generalize [17, Theorem T] in the sense thatfor largep, the freep-rank of a finite connected CW-complex with abelian fundamental group isbounded above by the rational homotopy Euler characteristic ofX. However, we will not discussthis in greater detail.

We believe that the methods developed in this paper are not sufficient to establish the generalform of Conjecture 1.2 for small primes. This problem remains open.

4 BERNHARD HANKE

Acknowledgements. I am grateful to Bill Browder for drawing my attention to [28], to JonathanBowden for proofreading a previous version of this manuscript and to the referees for numeroushelpful comments and remarks.

2. PROOF OFTHEOREM 1.3

Letp be a prime and assumeG := (Z/p)r acts freely onX = Sn1×. . .×Snk . Denoting byke thenumber of even dimensional spheres inX, we assume that the dimensionsn1, . . . , nke are even (andgreater than0) andnke+1, . . . , nk are odd. BecauseG acts freely, the Borel spaceXG := EG×GXis homotopy equivalent to the orbit spaceX/G. This is a topological manifold of dimensiondimXand consequentlyHn(XG;Fp) = 0 for all n > dimX. Assume thatp > 3 dimX. In particular,pis odd.

Under these assumptions the following result will be proved at the end of Section 5.

Proposition 2.1. There are free associative, commutative graded differential algebras with unit(E∗, dE) and(M∗, dM) overFp with the following properties:

1) as graded algebrasE∗ = Fp[t1, . . . , tr]⊗Λ(s1, . . . , sr)⊗M∗, where eachti is of degree2 andeachsi is of degree1.

2) the differentialdE is zero onFp[t1, . . . , tr]⊗Λ(s1, . . . , sr)⊗1 and the mapE∗ →M∗, ti, si 7→ 0,is a cochain map.

3) M∗ is the tensor product of commutative graded differential algebras(M∗j , dMj

), 1 ≤ j ≤ k,that are defined as follows:a) M∗

j = Λ(σj) with deg(σj) = nj anddMj(σj) = 0, if j > ke (i.e. if nj is odd),

b) M∗j = Fp[τj] with deg(τj) = nj anddMj

(τj) = 0, if j ≤ ke and2nj − 1 > dimX + 1 ,c) M∗

j = Fp[τj]⊗Λ(ηj) with deg(τj) = nj, deg(ηj) = 2nj−1 anddMj(τj) = 0, dMj

(ηj) = τ 2j ,

if j ≤ ke and2nj − 1 ≤ dimX + 1,4) each monomial int1, . . . , tr of cohomological degree at leastdimX + 1 represents the zero

class inH∗(E).

The short exact sequence ofFp-cochain algebras

Fp[t1, . . . , tr]⊗ Λ(s1, . . . , sr)→ E∗ →M∗

models the Borel fibrationX ↪→ XG → BG up to degreedimX + 1 (respectively up to de-gree dimX + 2 as far as the monomials int1, . . . , tr are concerned) implying the last state-ment of Proposition 2.1. In rational homotopy theory the Grivel-Halperin-Thomas theorem [4,Theorem 2.5.1] yields a similar short exact sequence of rational differential graded algebras forspaces equipped with(S1)r-actions that models the Borel fibration in all degrees. In this caseFp[t1, . . . , tr]⊗Λ(s1, . . . , sr) is replaced by a polynomial algebraQ[t1, . . . , tr], the minimal modelof B(S1)r, and the algebraM∗ can be chosen as the minimal model ofX. If X is a product ofspheres as before, this minimal model looks very similar to the algebraM∗ appearing in Proposi-tion 2.1. In Section 5 tame homotopy theory will be used to establish a counterpart to the Grivel-Halperin-Thomas theorem forG-spaces and cohomology withFp-coefficients which models theBorel fibration up to degreedimX + 1 and leads to the statement of Proposition 2.1. In the de-scription ofM∗ the two cases2nj − 1 > dimX + 1 and 2nj − 1 ≤ dimX + 1 need to bedistinguished, because forp > 3 dimX the procedure of Section 5 constructs free generators ofthe algebraE∗ only up to degreedimX + 1. When the algebraE∗ has been simplified, we caneasily add generatorsηj of degrees2nj − 1 > dimX + 1, see below.


Starting from(E∗, dE) we shall construct a commutative graded differential algebra(F ∗, δF )that is easier to handle. Let

F ∗ := Fp[t1, . . . , tr]⊗M∗

be obtained fromE∗ by dividing out the ideal(s1, . . . , sr). The induced differential onF ∗ isdenoted bydF .

Inspired by the construction ofpure towersin [17], we deformdF to another differentialδF onF ∗ as follows:δF is a derivation that vanishes onFp[t1, . . . , tr, τ1, . . . , τke ] and satisfies

δF (σj) = π(dF (σj))

δF (ηj) = π(dF (ηj)) .

whereπ : F ∗ → F ∗ is the projection ontoFp[t1, . . . , tr, τ1, . . . , τke ] given by evaluating the odddegree generatorsσj, ηj at 0. It is easy to verify thatδ2

F = 0.For l ≥ 0 let Σl ⊂ F ∗ be theFp[t1, . . . , tr]-linear subspace generated by the monomials inM∗

containing exactlyl of the odd degree generatorsσj, ηj, in particularΣl = 0 for l > k by theanticommutativity of the product (recall thatp is odd). We setΣ+ :=

⊕l≥1 Σl. This is a nilpotent

ideal inF ∗.

Lemma 2.2. For all l ≥ 1 the differentialδF mapsΣl to Σl−1. Furthermore, the image ofδF − dFis contained inΣ+.

Proof. The first assertion holds by the definition and derivation property ofδF .The second assertion holds for the generatorsσj andηj, becauseim(id−π) ⊂ Σ+, it holds for

the generatorsti, becauseδF anddF send these elements to zero and it holds for the generatorsτj, because eachdF (τj) is of odd degree and therefore contained inΣ+. This implies the secondassertion in general, sinceΣ+ is an ideal inF ∗ andδF − dF is a derivation. �

The elementsti, 1 ≤ i ≤ r, andτj, 1 ≤ j ≤ ke, represent cocycles in(F ∗, δF ). Let [ti] and[τj]be the corresponding cohomology classes. The proof of Theorem 1.3 depends on the followingcrucial fact.

Proposition 2.3. The elements[ti] are nilpotent inH∗(F, δF ). The same holds for[τj], if 2nj−1 ≤dimX + 1.

Proof. As noted earlier each monomial int1, . . . , tr of cohomological degree at leastdimX + 1represents the zero class inH∗(E). Letm be such a monomial and writem = dE(c) for a cochainc ∈ E∗. We will show thatm is a coboundary in(F ∗, δF ) as well.

First, after dividing out the ideal(s1, . . . , sr), we get an analogous equationm = dF (c) for somec ∈ F ∗. By Lemma 2.2 we concludeδF (c) = m+ ω whereω ∈ Σ+. Let c1 be the component ofcin Σ1. Lemma 2.2 and the fact thatm ∈ Σ0 imply the equationδF (c1) = m. This shows thatm isa coboundary in(F ∗, δF ).

In particular we have shown that the classes[ti] ∈ H∗(F, δF ) are nilpotent.The cochain algebra(F ∗, δF ) has a decreasing filtration given by

Fγ(F ∗) = Fp[t1, . . . , tr]≥γ ⊗M∗

whereγ ∈ N denotes the cohomological degree. Our previous argument and the fact that eachτjis a cocycle in(F ∗, δF ) imply that each element inΣ0 ⊂ F ∗ in filtration level at leastdimX + 1is a coboundary in(F ∗, δF ).

6 BERNHARD HANKE

Now pick aj ∈ {1, . . . , ke} with 2nj − 1 ≤ dimX + 1. Recall (see the description ofE∗ inProposition 2.1) that

dF (ηj) = τ 2j mod F2(F ) .

By the definition ofδF we obtain

δF (ηj) = π(τ 2j ) = τ 2

j mod F2(F ) ,

since the mapπ preserves the ideal(t1, . . . , tr) = F2(F ). This implies thatτ 2j is δF -cohomologous

to a cocyclec ∈ F2(F ∗, δF ). Hence,(τ 2j )dimX is δF -cohomologous tocdimX ∈ F2 dimX(F ). We

can splitcdimX into a sumc0 + c+ wherec0 ∈ Σ0 ∩ F2 dimX(F ) andc+ ∈ Σ+ ∩ F2 dimX(F ). Asnoted earlier,c0 is δF -cohomologous to zero. BecauseΣ+ is nilpotent, the elementc+ is nilpotent.

We conclude thatτ 2 dimXj is δF -cohomologous to a nilpotent cocycle in(F ∗, δF ). �

After these preparations we can finish the proof of Theorem 1.3. At first we adjoin new exteriorgeneratorsηj of degree2nj − 1 to F ∗ for all j ∈ {1, . . . , ke} satisfying2nj − 1 > dimX + 1.The differentialδF is extended to a differential on this new graded commutative algebra by settingδF (ηj) := τ 2

j , if 2nj − 1 > dimX + 1. This is possible, becauseδF (τj) = 0 by construction ofδF . This new commutative graded differential algebra is still denoted by(F ∗, δF ). Together withProposition 2.3 we see that the elementsti, 1 ≤ i ≤ r, andτj, 1 ≤ j ≤ ke, define nilpotent classesin H∗(F, δF ). This implies thatH∗(F, δF ) is a finite dimensionalFp-vector space.

We consider the ideal

I =(δF (η1), . . . , δF (ηke), δF (σke+1), . . . , δF (σk)

)⊂ Fp[t1, . . . , tr, τ1, . . . , τke ]

contained inim(δF ) and obtain an inclusion

Fp[t1, . . . , tr, τ1, . . . , τke ]/I ⊂ H∗(F, δF ) .

Here we use the fact that the coboundaries in(F ∗, δF ) are contained in the idealI · F ∗, whoseintersection withFp[t1, . . . , tr, τ1, . . . , τke ] is equal toI. This can be seen by applying the evalua-tion mapπ. We conclude thatFp[t1, . . . , tr, τ1, . . . , τke ]/I is a finite dimensionalFp-vector space.BecauseI is generated by homogenous elements of positive degree, it does not contain a unit ofFp[t1, . . . , tr, τ1, . . . , τke ] and hence there is a minimal prime idealp ⊂ Fp[t1, . . . , tr, τ1, . . . , τke ]containingI. The quotientFp[t1, . . . , tr, τ1, . . . , τke ]/p is both a finite dimensionalFp-vector spaceand an integral domain. Hencep = (t1, . . . , tr, τ1, . . . , τke) and consequentlyheight(p) = r + ke.By Krull’s Principal Ideal Theorem, see [15, Theorem 10.2], the number of generators ofI mustbe at leastr+ ke. From the definition ofI we derive the inequalityke + ko ≥ r+ ke. This impliesko ≥ r and finishes the proof of Theorem 1.3.

3. TAME HOMOTOPY THEORY VIA DIFFERENTIAL FORMS

The remainder of our paper is devoted to the construction of small approximativeFp-cochainmodels of Borel spaces associated to(Z/p)r-spaces in Theorem 5.6. Then the correspondingcochain models for(Z/p)r-actions on products of spheres will lead directly to the cochain algebras(E∗, dE) and(M∗, dM) appearing in Proposition 2.1.

First, in this section, we collect some fundamental notions and constructions from tame homo-topy theory via polynomial forms as initiated in [12] and further developed in [5, 21, 24, 26, 28, 30].As in the rational situation [6] it is convenient to formulate the theory in the language of simplicialsets. The reader unfamiliar with this may consult the introductory text [22].


Prime numbers are henceforth denoted by the letterl. Forq ∈ N we set

Qq := Z[l−1 | l ≤ q] ,

i.e.Qq is the smallest subring ofQ where all primesl ≤ q are invertible. In particular,Q0 = Q1 =Z.Definition 3.1. A filtered cochain complexis a cochain complex(V ∗, d) overQ, graded overN,together with an increasing sequence of subcomplexes

V ∗,0 ⊂ V ∗,1 ⊂ V ∗,2 ⊂ . . . ⊂ V ∗

so thatV ∗,q is a cochain complex ofQq-modules. We callV ∗,q the subcomplex offiltration levelq. A filtered graded differential algebrais a filtered cochain complex(A∗, d) which is equippedwith an associative multiplication with unit1 ∈ A0,0, which restricts to mapsA∗,q ⊗ A∗,q

′ →A∗,q+q

′. Furthermore,d is assumed to act as a graded derivation. If the multiplication is graded

commutative, thenA∗ is called afiltered commutative graded differential algebraor filtered CGDA.If A∗ is a filtered cochain complex or filtered graded differential algebra andx ∈ Ap, we callp

thedegreeof x andmin{q |x ∈ Ap,q} thefiltration degreeof x. The category of filtered CGDAsand filtration preserving CGDA maps is denoted byA. It can be equipped with the structure of aclosed model category [21, 25, 26, 28, 30], but this fact will not be needed for our discussion.

Let V1 andV2 be two filtered cochain complexes. Theirtensor productis the cochain complexV1 ⊗ V2 with the usual grading and differential together with the filtration

(V1 ⊗ V2)q :=∑

q1+q2=q

V ∗,q11 ⊗ V ∗,q22 ⊗Qq ⊂ V ∗1 ⊗ V ∗2 ,

where∑

denotes a sum which need not be direct. The same definition applies to filtered CGDAs.The tensor product defines the coproduct inA.

If V is a filtered cochain complex, letΛV denote the free CGDA generated byV . It is equippedwith a filtration as described in the preceding paragraph. Note that this is an algebra with unit1 ∈ (ΛV )0,0 and that the filtration degree of a monomialv1 · . . . · vn, vi ∈ V , is less than or equalto the sum of the filtration degrees of thevi.

Sometimes the following variaton of the tensor product will be useful. Thefiltrationwise tensorproductV1⊗V2 is defined as the tensor product complexV1 ⊗ V2, but equipped with the filtration

(V1⊗V2)∗,q := V ∗,q1 ⊗ V ∗,q2 .

The prototypical example of a filtered CGDA is provided by the cubical cochain algebra of Cenkland Porter [12] which refines the Sullivan-de Rham algebra [6,§1] used in rational homotopytheory. Whereas the latter is based on forms on the standard simplices∆n ⊂ Rn+1, the Cenkl-Porter construction works with forms on a cubical decomposition of∆n. This is essential fordefining a filtration on the resulting cochain algebra in such a way that in each positive filtrationlevel one gets a cohomology theory in the sense of Cartan [11]. We recall the essential steps of thisconstruction.

We cubically decompose the standardn-simplex by regarding it as the subset

∆n := {(x0, . . . , xn) ∈ Rn+1 | 0 ≤ xi ≤ 1,n∏i=0

xi = 0}

of the boundary of the(n+ 1)-dimensional cube. The verticesv0, . . . , vn of ∆n are given by

vi := (1, 1, . . . , 1, 0, 1, . . . , 1)

8 BERNHARD HANKE

with 0 located at thei-th entry,0 ≤ i ≤ n. The inclusion of thei-th face into∆n and thei-thcollapse onto∆n (0 ≤ i ≤ n) are the maps

δi : ∆n−1 → ∆n , (x0, . . . , xn−1) 7→ (x0, . . . , xi−1, 1, xi+1, . . . , xn−1)

σi : ∆n+1 → ∆n , (x0, . . . , xn+1) 7→ (x0, . . . , xi−1, xi · xi+1, xi+2, . . . , xn+1) .

We consider the free commutative graded algebra overZ

Z[t0, . . . , tn]⊗ ΛZ(dt0, . . . , dtn)

with generatorst0, . . . , tn in degree0 anddt1, . . . , dtn in degree1. To a monomial

tα00 dt

ε00 · . . . · tαnn dtεnn , αi ≥ 0 , 0 ≤ εi ≤ 1 ,

in this algebra we assign thefiltration degreemaxi{αi + εi}. The monomials of filtration degree0 are exactly the constant ones. LetI be the homogenous ideal generated by the monomials withαi + εi > 0 for all i. The quotient

Z[t0, . . . , tn]⊗ ΛZ(dt0, . . . , dtn)/I

is again a graded algebra and is called thealgebra of compatible polynomial forms on the cubicaldecomposition of∆n.

Forp, q ≥ 0, we define

T p,qn (Z) ⊂ (Z[t0, . . . , tn]⊗ ΛZ(dt0, . . . , dtn)/I)p

as theZ-module generated by the (cosets of) monomials of degreep and filtration degree at mostq. Exterior product of polynomial forms induces maps

∧ : T p1,q1n (Z)⊗ T p2,q2

n (Z)→ T p1+p2,q1+q2n (Z) .

We define a coboundary mapd : T p,qn (Z)→ T p+1,q

n (Z)

by xi 7→ dxi, dxi 7→ 0 and extend it to the whole ofT p,qn (Z) by the Leibniz rule. Upon defining

T p,qn := T p,qn (Z)⊗Qq

we obtain a filtered CGDAT ∗n for eachn ≥ 0. Via pullback of forms, the mapsδi andσi inducemaps of filtered CGDAs

∂i : T ∗n → T ∗n−1

si : T ∗n → T ∗n+1

that satisfy the simplicial identities. In other words,(T ∗n)n∈N is a simplicial filtered CGDA.Let S denote the category of simplicial sets. For any simplicial setX ∈ S, we define a filtered

CGDAT ∗,q(X) := MorS(X,T ∗,q) ,

which is called theCenkl-Porter complex of polynomial forms onX. Note that by definitionT ∗,0(X) is concentrated in degree0 and can be identified withMap(π0(X),Z). Conversely, ifA ∈ A is a filtered CGDA, itssimplicial realizationis the simplicial set‖A‖ defined by

‖A‖n := MorA(A, T ∗n)

for n ≥ 0. Note that corresponding constructions exist in rational homotopy theory, see [6].For a proof of the following almost tautological statement, see [6, 8.1].


Proposition 3.2. LetX be a simplicial set andA be a filtered CGDA. Then there is a canonicalbijection

MorS(X, ‖A‖) ≈ MorA(A, T (X))

which is natural inA andX. In other words the simplicial realization‖ − ‖ and the Cenkl-PorterfunctorT (−) define a pair of right adjoint contravariant functors.

Let us fix the notationΨA : A 7→ T (‖A‖) for the unit defined by this adjunction.We will now compareH∗(T ∗,q(X)) with the simplicial cohomology ofX. This can be done

most in the axiomatic setting described in [11]. We call a simplicial cochain complex(A∗n)n∈Nover a commutative ringR with identity acohomology theory, if the sequence of simplicialR-modules

A0 d→ A1 d→ A2 → . . .

is exact, if the kernelZA0 of the differentiald : A0 → A1 is simplicially trivial, i.e. all face anddegeneracy maps are isomorphisms, and if for eachp ≥ 0, the simplicial abelian groupAp iscontractible, cf. [11, Section 2]. Concerning the latter property, recall [22, Theorem 22.1] that fora simplicial abelian group(Gn)n∈N, thep-th homotopy groupπp(G) can be computed as thep-thhomology of the chain complex

. . .→ Gn → Gn−1 → Gn−2 → . . .

with differentials equal to the alternating sum of the face operators ofG. Given a cohomologytheoryA∗, we call theR-moduleR(A) := (ZA0)0 thecoefficientsof A. Furthermore we get anR-cochain complex

A∗(X) := MorS(X,A∗)

whose cohomology is naturally isomorphic toH∗(X;R(A)), the simplicial cohomology ofX withcoefficients inR(A), see [11, Theoreme 2.1].

Proposition 3.3. For eachq ≥ 1, the simplicial CGDAT ∗,q defines a cohomology theory withcoefficientsQq.

Proof. The sequence

T 0,q d→ T 1,q d→ . . .

of simplicialQq-modules is exact by the cubical Poincare lemma [12, Proposition 3.5]. The con-tractibility of the simplicial setsT p,q for all p ≥ 0 follows from the cubical extension lemma [12,Lemma 3.1]. That the kernel of the differentialT 0,q → T 1,q is simplicially trivial is obvious (itconsists of constant monomials). �

It is exactly at this point, where division by numbers smaller than or equal toq is required:for 2 ≤ k ≤ q the closed formtk−1dt lives in filtration levelq and is the coboundary oftk/k.Furthermore, notice thatT ∗,0 is not a cohomology theory. IndeedT 0,0 → T 1,0 → T 2,0 → . . . isexact andT 0,0 is simplicially trivial, butT 0,0 is not connected.

The last proposition implies that ifq ≥ 1 andM is aQq-module, thenM⊗T ∗,q is a cohomologytheory with coefficientsM . This follows from the universal coefficient theorem and the fact thatT ∗,q consists of torsion free and hence flatQq-modules, cf. [29, Theorem 5.3.14].

Corollary 3.4. Let q ≥ 1 and letM be aQq-module. Then, for anyp ≥ 0, the simplicialQq-moduleM ⊗ T p,q is contractible and for anyp ≥ 1, the simplicialQq-moduleM ⊗ ZT p,q is anEilenberg-MacLane complex of type(M, p).

10 BERNHARD HANKE

Proof. Both statements are implied by the fact thatM ⊗ T ∗,q is a cohomology theory. The firststatement is then immediate. The second statement is proved by an inductive argument as in [11,Theoreme 1] combined with the equationM ⊗ ZT p,q = ker(id⊗d) : M ⊗ T p,q → M ⊗ T p+1,q

which follows from the universal coefficient theorem applied to the cochain complex of flatQq-modules0→ T p,q → T p+1,q → T p+2,q → . . . �

In order to study the relation ofH∗(T ∗,q(X)) andH∗(X;Qq), we first introduce for eachq ≥ 1the cohomology theory(C∗,qn )n∈N, cf. [11, Section 3], with coefficientsQq, whereC∗,qn is the sim-plicial cochain complexC∗(∆[n];Qq) of the simplicialn-simplex∆[n]. For each simplicial setX,we will identify C∗,q(X) with C∗(X;Qq), the simplicial cochain complex ofX with coefficientsin Qq.

For q ≥ 1 we introduce a third simplicial cochain complex((T ⊗C)∗,qn )n∈N over Qq with(T ⊗C)∗,qn := T ∗,qn ⊗ C∗,qn and with face and degeneracy maps which are the tensor productsof the corresponding maps onT ∗,qn andC∗,qn . The Kunneth formula [29, Corollary 5.3.4] and theEilenberg-Zilber theorem [20, Theorem 8.1] imply that(T ⊗C)∗,q is again a cohomology theorywith coefficientsQq. We have canonical mapsT ∗,q → (T ⊗C)∗,q andC∗,q → (T ⊗C)∗,q ofsimplicial cochain complexes given by

T ∗,qn = T ∗,qn ⊗ (Qq · 1) ↪→ T ∗,qn ⊗ C∗,qnC∗,qn = (Qq · 1)⊗ C∗,qn ↪→ T ∗,qn ⊗ C∗,qn .

These induce isomorphisms of coefficients and hence [11, Proposition 2] we get induced isomor-phisms

H∗(T ∗,q(X)) ∼= H∗((T ⊗C)∗,q(X))

H∗(C∗,q(X)) ∼= H∗((T ⊗C)∗,q(X)) ,

which are natural inX, for eachq ≥ 1. Now we observe that exterior multiplication of cubicalforms and the cup product of simplicial cochains define maps of simplicial cochain complexes

T ∗,q1n ⊗ T ∗,q2n → T ∗,q1+q2n

C∗,q1n ⊗ C∗,q2n → C∗,q1+q2n

(T ⊗C)∗,q1n ⊗ (T ⊗C)∗,q2n → (T ⊗C)∗,q1+q2n

which are compatible with the simplicial cochain mapsT ∗,qn → (T ⊗C)∗,qn andC∗,qn → (T ⊗C)∗,qnconsidered before. We hence obtainTheorem 3.5(Cenkl-Porter theorem [12]). For q ≥ 1 there are isomorphisms ofQq-modules

H∗(T ∗,q(X)) ∼= H∗(X;Qq)

which are natural inX. These are compatible with the multiplicative structures in the sense that

Hp1(T ∗,q1(X))⊗Hp2(T ∗,q2(X))∧ //

∼=��

Hp1+p2(T ∗,q1+q2(X))

∼=��

Hp1(X;Qq−1)⊗Hp2(X;Qq2)∪ // Hp1+p2(X;Qq1+q2)

commutes for allq1, q2 ≥ 1 and multiplicative units are mapped to each other.Observe that the isomorphismH∗(T ∗,q(X)) ∼= H∗(X;Qq) is induced by the zig-zag sequence

of filtered graded differential algebras

(3.6) T ∗(X)→ (T ⊗C)∗(X)← C∗(X)


where both arrows induce isomorphisms in cohomology. In contrast, the proof of Theorem 3.5 in[12] is based on a single map of filtered cochain complexes∫

: T ∗,q(X)→ C∗(X,Qq)

induced by integration of forms. One can indeed show that∫

induces an isomorphism of cohomol-ogy groups which is compatible with multiplicative structures. However,

∫is not multiplicative

on the cochain level. Using the uniqueness of natural transformations of cohomology theories [11,Proposition 2] one can show that the isomorphisms induced by

∫and the one in our Theorem 3.5

coincide.Concerning filtration level zero, we note the existence of a canonical isomorphism

H∗(T ∗,0(X)) ∼= H0(X;Z) .

Using this the multiplicativity property in Theorem 3.5 holds forq1, q2 ≥ 0.As an addendum to the Cenkl-Porter theorem, we remark that for anyq ≥ 1 and for any prime

l > q, we have isomorphisms

H∗(T ∗,q(X)⊗ Fl) ∼= H∗(X;Fl) ,

which are multiplicative for filtration levelsq1, q2 ≥ 1 with l > q1 + q2. This is shown by tensoringthe cochain complexes appearing in (3.6) withFl and applying the universal coefficient theorem.

We finish this section with a discussion of higher Massey products. Letl be an odd prime. Itis well known [19] that thel-fold Massey productH1(BZ/l;Fl) → H2(BZ/l;Fl) is nonzero.Because squares of elements of degree one in commutative cochain algebras are equal to zero,one might wonder if this nonvanishing result does not contradict the Cenkl-Porter theorem and inparticular the addendum stated above. The following discussion will clarify this point.

First we recall that Massey products are defined on singular cochain algebrasC∗(X;R) of topo-logical spacesX, respectively simplicial cochain complexes of simplicial setsX, whereR is somecommutative ring with unit. Because our proof of Theorem 3.5 is based on (3.6) in which all mapsare multiplicative on the cochain level, we can use the filtered cochain algebraT ∗,∗(X) or - in viewof the addendum to the Cenkl-Porter theorem - the complexesT ∗,q(X) ⊗ Fl to construct higherMassey products inH∗(X;Fl). But now the filtration structure ofT ∗,∗(X) has to be considered.More precisely, when constructing anl-fold Massey product of a cochainc ∈ T 1,1(X) ⊗ Fl, wehave to performl − 1 multiplications (cf. the construction of Massey products in [19]) and hencearrive at elements inT ∗,l(X)⊗ Fl. But this group is zero, becausel is invertible inT ∗,l(X).

This reasoning indicates that it is exactly the filtration structure together with its divisibilityproperties that allows us to generalize the construction of the Sullivan-de Rham algebra to an non-rational setting.

4. TAME HIRSCH LEMMA

In this section we shall explain a technique that will enable us to replace the Cenkl-Porter com-plex by a smaller filtered CGDA which reflects the topological structure of the underlying simpli-cial set in a similar manner as the minimal model does in rational homotopy theory.

Of fundamental importance is thefiltration function· : N+ → N+ defined by

1 = 1 , t = 3(t− 1) for t ≥ 2 .

12 BERNHARD HANKE

One easily checks the followingadmissibilityproperty: lett1, . . . , tn ≥ 1 andn ≥ 2 be naturalnumbers. Ifn = 2, let t1, t2 ≥ 2. Then the following implication holds for eacht ∈ N+:

n∑i=1

ti ≤ t+ 1 =⇒n∑i=1

ti ≤ t .

It is easy to see that there is no functionN+ → N+ with this property without this additionalassumption ifn = 2. Futhermore, the above filtration function is optimal in the sense that anyother admissible filtration function is pointwise greater than or equal to the one described above.

In the literature on tame homotopy theory (see e.g. [14, 26]), homotopy, homology and coho-mology groups are indexed by degreesr+ k, where(r− 1) denotes the connectivity of the spacesunder consideration, and filtration functions depend on the parametert = k. In our case we haver = 1, and in order to simplify the notation, we use the definition above which expresses thefiltration function in terms of the actual degreet = r + k.

Definition 4.1. Let f : A∗ → B∗ be a map of filtered CGDAs. The mapf is called aprimaryweak equivalence, if for all primesl > q the mapf ⊗ id : A∗,q⊗Fl → B∗,q⊗Fl is at-equivalencefor all t ≥ 1 andq ≥ t, i.e. the induced map in cohomology is an isomorphism in degrees less thanor equal tot and injective in degreet+ 1. .

Given a natural numberk ≥ 1, we say that a primary weak equivalencef : A∗ → B∗ satisfiesconditionk+, if for all primesl > q the mapf⊗ id : A∗,q⊗Fl → B∗,q⊗Fl is a(t+1)-equivalencefor all t ≥ k andq ≥ max{t, k + 1}.

Note the close relationship between filtration level and cohomological degree. The “extra de-gree” in conditionk+ is essential for the inductive step in the construction of a small filteredcochain model along a Postnikov decomposition of the underlying simplicial set, see the proof ofLemma 5.5.

Definition 4.2. Let (B, dB) be a filtered CGDA,(V, dV ) a filtered cochain complex andτ : V ∗ →B∗+1 a filtration preserving cochain map of degree1, i.e.τ satisfies the equationdB ◦τ = −τ ◦dV .The free extension ofB by V with twistingτ , denotedB ⊗τ ΛV , is the filtered graded algebraB⊗ΛV equipped with the unique differentiald which satisfiesd(b⊗ 1) = dB(b)⊗ 1 for all b ∈ Bandd(1⊗ v) = τ(v)⊗ 1 + 1⊗ dV (v) for all v ∈ V and which acts as a graded derivation.

Of particular importance are free extensions by so called elementary complexes.

Definition 4.3. Let (p, q) be a pair of natural numbers withp, q ≥ 1. An elementary complexoftype(p, q) is a filtered cochain complex(V ∗, η) such that

V ∗,q′=

{V ∗,q ⊗Qq′ for q′ ≥ q

0 for 0 ≤ q′ < q ,

andV ∗,q is of the form. . .→ 0→ V p,q η→ V p+1,q → 0→ . . .

whereV p,q andV p+1,q are finitely generated freeQq-modules and the cokernel ofη is a torsionmodule.

If V is an elementary complex of type(p, q), then we let

η′ : (V p+1,q)′ := Hom(V p+1,q,Qq)→ (V p,q)′

denote the dual ofη. By assumption the mapη′ is injective. We will show that simplicial realiza-tions of elementary complexes are Eilenberg-MacLane complexes for the groupcoker η′. To this


end let us first transfer some elementary constructions involving cochain complexes to the filteredsetting.

If f : (V, dV ) → (W, dW ) is a map of filtered cochain complexes, we denote by(Cf , dCf ) themapping coneof f , i.e. Cp,q

f := V p,q ⊕ W p−1,q, where we setW−1,q = 0, anddCf (v, w) :=(dV (v), f(v)− dW (w)). Special cases are thesuspensionΣW of W , defined as the mapping coneof 0 → W , and theconeoverV , defined as the mapping cone ofid : V → V and denoted byconeV . Note that we have a canonical short exact sequence

0→ ΣW → Cf → V → 0

of filtered cochain complexes.For the notions appearing in the following proposition and its proof see [22].

Proposition 4.4. Let (V, η) be an elementary complex of type(p, q). Then‖ΛV ‖ is a (not neces-sarily minimal) Eilenberg-MacLane complex of type(coker η′, p). In particular, it satisfies the Kancondition.

Proof. By definition the set ofn-simplices of‖ΛV ‖ is given by

‖ΛV ‖n = MorA(ΛV, Tn) = MorC(V, Tn) ,

whereC denotes the category of filtered cochain complexes. We conclude that‖ΛV ‖ is a simplicialabelian group, thus fulfilling the Kan condition, that fits into a pullback square of simplicial abeliangroups

‖ΛV ‖ //

��

(V p,q)′ ⊗ T p,q

id⊗d��

(V p+1,q)′ ⊗ ZT p+1,q η′⊗id // (V p,q)′ ⊗ ZT p+1,q

By Corollary 3.4,(V p,q)′⊗ZT p+1,q is an Eilenberg-MacLane complex of type((V p,q)′, p+ 1) and(V p,q)′ ⊗ T p,q is contractible. Furthermore, the right hand vertical map is surjective with kernel(V p,q)′ ⊗ ZT p,q. It is hence a principal Kan fibration with contractible total space and fibre anEilenberg-MacLane complex of type((V p,q)′, p), cf. [22, Lemma 18.2]. The long exact homotopysequence for the induced fibration‖ΛV ‖ → (V p+1,q)′⊗ZT p+1,q shows that‖ΛV ‖ is an Eilenberg-MacLane complex of type(coker η′, p). �

Given an elementary complex(V, η) of type (p, q) we will show that the simplicial realizationfunctor transforms the sequence

ΣV → coneV → V

into a fibre sequence‖ΛV ‖ ↪→ ‖Λ(coneV )‖ → ‖Λ(ΣV )‖ .

For this we note that the map of simplicial abelian groups‖Λ(coneV )‖ → ‖Λ(ΣV )‖ is sur-jective and hence a principal Kan fibration, whose kernel can be identified with the simplicialabelian group‖ΛV ‖. The last point can be checked using an explicit description of‖Λ(coneV )‖and‖Λ(ΣV )‖ similar to the one of‖ΛV ‖ in the proof of Proposition 4.4. This explicit descrip-tion also shows that the homotopy groups of‖Λ(coneV )‖ are zero in all degrees and hence that‖Λ(coneV )‖ is contractible. By Proposition 4.4 the simplicial set‖Λ(ΣV )‖ is an Eilenberg-MacLane complex of type(coker η′, p+ 1). We conclude that the fibration

‖ΛV ‖ ↪→ ‖Λ(coneV )‖ → ‖Λ(ΣV )‖

14 BERNHARD HANKE

is fibre homotopy equivalent to the path-loop fibration over an Eilenberg-MacLane complex oftype(coker η′, p + 1). Fibrations of this kind can thus be used as building block in the Postnikovdecompositions of simplicial sets.

Proposition 4.5. LetV be an elementary complex of type(p, q) with V p,q = Qq andV p+1,q = 0.Then the unitΨΛV : ΛV → T (‖ΛV ‖) induces an isomorphismHp((ΛV )∗,q)→ Hp(T ∗,q(‖ΛV ‖))of groups that can be canonically identified withQq.

Proof. The generator ofHp(T ∗,q(ZT p,q)) = Qq is represented by the inclusion mapZT p,q ↪→ T p,q

in MorS(ZT p,q, T p,q). Now recall from the proof of Proposition 4.4 that‖ΛV ‖ = (V p,q)′⊗ZT p,q.We thus obtain the unit

ΨΛV : (ΛV )p,q → MorS((V p,q)′ ⊗ ZT p,q, T p,q

)and this sendsc ∈ (ΛV )p,q = Qq to the morphism of simplicial sets(V p,q)′ ⊗ ZT p,q → T p,q,φ⊗ zn 7→ φ(c) · zn. Together with the previous remark this implies the assertion. �

Now letX be a simplicial set, letV be an elementary complex and let

f : X → ‖Λ(ΣV )‖be a simplicial map. We obtain a pull back square

Ef //

pf

��

‖Λ(coneV )‖

��

Xf // ‖Λ(ΣV )‖

The adjoint off is a map of filtered CGDAsΛ(ΣV ) → T (X) and this is induced by a filteredcochain mapf ] : V → T (X) of degree1. We hence get a free extensionT (X) ⊗f] ΛV and themapsT (pf ) : T (X)→ T (Ef ) and the compositionV ↪→ coneV → T (Ef ) induced by the adjointof Ef → ‖Λ(coneV )‖ induce a filtered CGDA map

Γf : T (X)⊗f] ΛV → T (Ef ) .

The following result will be important in the next section. It is a combination of [28, TheoremII.6.4.(i)] and [28, Lemma II.8.0]. A detailed proof will be given in Section 6.

Theorem 4.6(Tame Hirsch lemma). Let V be an elementary complex of type(d, k), whered ≥k ≥ 1, letX be a simplicial set and letf : X → ‖Λ(ΣV )‖ be a simplicial map. Then the inducedmap

Γf : T (X)⊗f] ΛV → T (Ef )

is a primary weak equivalence satisfying conditionk+.

We conclude this section with an example which illustrates a fundamental problem with ele-ments of degree1 and justifies the specific form of Definition 4.1. LetV be an elementary com-plex of type(1, 1), concentrated in degree1 with V 1,1 = Z. Then‖ΛV ‖ is an Eilenberg-Mac Lanecomplex of type(Z, 1) and for eachq ≥ 1, the unit(ΛV )∗,q → T ∗,q(‖ΛV ‖) induces isomorphismsin cohomology in all degrees. For a constant mapf : ‖ΛV ‖ → ‖Λ(ΣV )‖, we study the map

Γf : T (‖ΛV ‖)⊗ ΛV → T (‖ΛV ‖ × ‖ΛV ‖)that appears in the tame Hirsch lemma. This map is not a primary2-equivalence in filtration level1, becauseH2((T (‖ΛV ‖)⊗ΛV )∗,1) = 0, butH2(T ∗,1(‖ΛV ‖× ‖ΛV ‖)) = Z by the Cenkl-Porter


theorem. This example shows that the extra degree in propertyk+ does not occur below filtrationlevelk + 1 in general.

5. SMALL COCHAIN MODELS FOR l-TAME (Z/l)r-SPACES

Let l be a prime,G := (Z/l)r andX a G-space. Under certain assumptions onX we shallconstruct a small commutative graded differential algebra overFl which is free as a graded algebraand whose cohomology is equal toH∗(XG;Fl) up to a certain degree. In particular we will proveProposition 2.1 from Section 2.

From now on it will be convenient to work exclusively in anl-local setting. This means thatall filtered cochain complexes and filtered cochain algebras appearing in Sections 3 and 4 will betensored withZ(l) in each filtration level. Since tensoring withZ(l) is exact, we still obtain filteredobjects. For example, ifV is a filtered cochain complex, then it is understood from now on thatV ∗,q is a complex ofZ(l)-modules for0 ≤ q < l and a complex ofQ-modules forq ≥ l. TheCenkl-Porter theorem now states that we get natural multiplicative isomorphisms

H∗(T ∗,q(X)) ∼=

{H∗(X;Z(l)) for 1 ≤ q < l

H∗(X;Q) for q ≥ l ,

where filtration levels are additive under multiplication of elements. Of course, in the definition anddiscussion of primary weak equivalences satisfying conditionk+, one now restricts to the singleprime l and to filtration levelsq < l. This is underlined by the notionl-primary weak equivalencesatisfying conditionk+, which will be used in the sequel.

Definition 5.1. We callX anl-tameG-space, ifX is path connected,π1(X) is abelian and for allt ≥ 1 with t < l the following holds:

1) πt(X) is a finitely generated freeZ(l)-module,2) π1(X) acts trivially onπt(X) and3) the inducedG-action onπt(X) is trivial.

Lemma 5.2. LetX be anl-tameG-space. Thenπ1(XG) is abelian and for eacht ≥ 2 with t < l,πt(XG) is a finitely generated freeZ(l)-module on whichπ1(XG) acts trivially.

Proof. We only show thatπ1(XG) is abelian. The proofs of the other assertions are elementary andleft to the reader.

The short exact sequence1→ π1(X)→ π1(XG)→ G→ 1 can be used to construct a map

φ : G×G→ π1(X) , (g1, g2) 7→ g1 · g2 · (g1)−1 · (g2)−1 .

Hereg1 andg2 are lifts ofg1, g2 ∈ G = (Z/l)r to π1(XG). The mapφ is well-defined because theconjugation action ofG onπ1(X) is trivial. Futhermore,φ is Z/l-bilinear. BecauseG is a torsiongroup andπ1(X) is a freeZ(l)-module,φ must be the zero map and henceπ1(XG) is abelian. �

Later we will need the following result, which is important to study maps between cochaincomplexes overZ(l).

Lemma 5.3. Let l be a prime and letV ∗ andW ∗ be (unfiltered) cochain complexes overZ(l) thatare torsion freeZ(l)-modules in each degree. Furthermore assume thatH∗(V ∗) andH∗(W ∗) arefinitely generatedZ(l)-modules in each degree.

Now letf : V ∗ → W ∗ be a map of cochain complexes so thatf ⊗ id : V ∗ ⊗ Fl → W ∗ ⊗ Fl is at-equivalence for somet ≥ 1. Thenf ∗ : V ∗ → W ∗ is itself at-equivalence.

16 BERNHARD HANKE

Proof. The short exact sequence

0→ ΣW ∗ → Cf → V ∗ → 0

induces long exact sequences in cohomology

. . .→ H i(Cf ⊗M)→ H i(V ⊗M)f∗→ H i(W ⊗M)→ . . .

for anyZ(l)-moduleM . Our assumptions imply thatH i(Cf ⊗ Fl) = 0 for all 0 ≤ i ≤ t + 1.Also,Cf consists of flatZ(l)-modules in each degree. Hence we can apply the universal coefficienttheorem [29, Theorem 5.3.14] to conclude thatH i(Cf ) ⊗ Fl = 0 and hence thatH i(Cf ) = 0 for0 ≤ i ≤ t + 1. Here we use thatH∗(Cf ) is finitely generated overZ(l) in each degree. From thisthe assertion follows. �

From now on we work in a simplicial setting and assume that we are given a Kan complexX equipped with aG-action. This can be achieved by passing to the simplicial set of singularsimplices of a givenG-space. As our model for the universal fibrationEG → BG we take theuniversal coveringEG → ‖ΛV0‖, whereV0 is the elementary complex of type(1, 1) given onfiltration level1 by the cochain complex

(5.4) 0→ Zr(l)

c 7→l·c→ Zr(l) → 0→ 0→ . . . ,

compare Proposition 4.4. In this way we realize the Borel fibrationX ↪→ XG → BG as a Kanfibration.

Now letX be anl-tameG-complex. We will study the Postnikov decomposition ofXG relativetoBG. This leads to a commutative diagram

�� X2

//

p2

��

(XG)2//

P2

��

BG

X1//

p1

��

(XG)1P1 //

P1

��

BG

X0 = ? // (XG)0 = BG BG

where for allk ≥ 1 the complexesXk and(XG)k arek-th stages of the Postnikov decompositionsof X andXG, each row is a fibre sequence with fibreXk and the vertical mapspk andPk are Kanfibrations whose fibres are Eilenberg-MacLane complexes of type(πk(X), k). By the assumptionthatX is l-tame together with Lemma 5.2, the mapsXk → Xk−1 and (XG)k → (XG)k−1 aresimple fibrations, ifk ≥ 1 andk < l, i.e the fundamental group of the base acts trivially on thek-th homotopy group of the fibre. Therefore, by Proposition 4.4 and the subsequent remarks, we


can assume that these maps fit into diagrams

‖Λ(coneVk)‖

��

‖Λ(coneVk)‖

��

Xk

77nnnnnnnn //

pk

��

(XG)k

66mmmmmmm

Pk

��

‖Λ(ΣVk)‖ ‖Λ(ΣVk)‖

Xk−1//

fk−1

77nnnnnnn(XG)k−1

Fk−1

66mmmmmmm

where(Vk, η) is an elementary complex of type(k, k) which is concentrated in degreek withV k,k = πk(X).

Fork = 0 and fork ≥ 1, k < l, we will define filtered CGDAsMk andEk together with filteredCGDA maps

φk : Ek → Mk

ψk : Mk → T (Xk)

Ψk : Ek → T ((XG)k)

fitting into commutative diagrams

T (Xk) T ((XG)k)oo

Mk

ψk99ssssssssss

Ek

Ψk

88rrrrrrrrrrrrφkoo

T (Xk−1)

T (pk)

OO

T ((XG)k−1)oo

T (Pk)

OO

Mk−1

ψk−1

99tttttttttt

OO

Ek−1

φk−1oo

Ψk−1

88rrrrrrrrrr

OO

Furthermore, fork ≥ 1 the mapsψk : Mk → T (Xk) and Ψk : Ek → T ((XG)k) will be l-primary weak equivalences satisfying conditionk+, the mapsψ0 andΨ0 will be l-primary weakequivalences satisfying condition1+ and the vertical maps in the front square will be inclusionswhich can be used to expressMk andEk as free extensions ofMk−1 andEk−1 making the followingdiagram commutative:

Mk Ekφkoo

Mk−1 ⊗φk−1◦τk ΛVk Ek−1 ⊗τk ΛVkφk−1⊗id

oo

We define these objects by induction onk. LetM∗,q0 = Z(l), concentrated in degree0, for 0 ≤ q <

l, and letM∗,q0 = Q, concentrated in degree0, for q ≥ l. Furthermore, defineE0 := ΛV0, where

18 BERNHARD HANKE

the cochain complexV0 was defined in (5.4). The maps

φ0 : E0 → M0

ψ0 : M0 → T (X0) = T (?)

Ψ0 : E0 → T ((XG)0) = T (‖V0‖)are the obvious ones:φ0 is an isomorphism in degree0 andΨ0 is the unit. The mapΨ0 is anl-primary weak equivalence satisfying condition1+ by Proposition 6.6. Now letk ≥ 1 andk < land assume that the above objects have been constructed fork − 1.

Lemma 5.5. For k ≥ 1 andk < l the mapΨk−1 induces an isomorphism

Ψ∗k−1 : Hk+1(E∗,kk−1) ∼= Hk+1(T ∗,k((XG)k−1)

).

Proof. At first we notice that the cochain complexesE∗,kk−1 andT ∗,k((XG)k−1) consist of torsion

freeZ(l)-modules in each degree. Moreover,H∗(E∗,kk−1) andH∗(T ∗,k((XG)k−1)) are finitely gen-erated overZ(l) in each degree. This holds by the Cenkl-Porter theorem and becauseπ∗((XG)k−1)

is finitely generated overZ(l) in each degree. Remember thatX is l-tame and thatk < l.Now, for k ≥ 2 the assertion follows by Lemma 5.3, becauseΨk−1 is anl-primary weak equiv-

alence satisfying condition(k − 1)+ and hence anl-primary(k + 1)-equivalence in filtration levelk.

Fork = 1 we getEk−1 = ΛV0 and the composition

(ΛV0)∗,1 ⊗ Fl → T ∗,1(‖ΛV0‖)⊗ Flis a1-equivalence by Proposition 6.6. An application of Lemma 5.3 shows that the induced map

H2((ΛV0)∗,1)→ H2(T ∗,1(‖ΛV0‖))is injective. It is hence an isomorphism as both source and target are isomorphic to(Z/l)r. �

Notice that in view of the remarks at the end of Section 4 it is somewhat unexpected that we stillget an isomorphism fork = 1.

BecauseH∗(ΣVk) is a freeZ(l)-module concentrated in degreek+ 1, Lemma 5.5 shows that themap

H∗((ΣVk)∗,k)→ H∗(Λ(ΣVk)

∗,k)→ H∗(T ∗,k((XG)k−1))

induced by the adjoint ofFk−1 can be written as the composition ofΨ∗k−1 and a map

H∗((ΣVk)∗,k)→ H∗(E∗,kk−1) .

This map is induced by a map of filtered cochain complexes

τk : ΣV ∗k → E∗k−1

and defines the free extensionsEk = Ek−1 ⊗τk ΛVk andMk = Mk−1 ⊗φk−1◦τk ΛVk as well as themapφk := (φk−1 ⊗ id) : Ek →Mk.

It remains to construct thel-primary weak equivalences satisfying conditionk+

ψk : Mk → T (Xk)

Ψk : Ek → T ((XG)k) .

LetF ]k−1 : ΣVk → T ((XG)k−1)


be the restriction of the adjoint ofFk−1.The composition

Ψk−1 ◦ τk : ΣV ∗,kk → T ∗,k((XG)k−1)

induces the same map in cohomology as the mapF ]k−1. We denote byFk−1 the adjoint ofΨk−1 ◦

Λ(τk) : Λ(ΣVk) → Ek−1 → T ((XG)k−1) and conclude that the mapsFk−1 andFk−1 are homo-topic as maps of simplicial sets(XG)k−1 → ‖Λ(ΣVk)‖. This follows, because‖Λ(ΣVk)‖ is anEilenberg-MacLane complex of type(πk(X), k + 1) and because after applying the functorT ∗,k

and precomposing with the unit

ΨΛ(ΣVk)|ΣV ∗,kk

: ΣV ∗,kk → T ∗,k(‖Λ(ΣVk)‖) ,

we precisely obtain the mapsF ]k−1 andΨk−1 ◦ τk considered at the beginning of this paragraph,

see also Proposition 4.5.By precomposing with the inclusionXk−1 → (XG)k−1 we also obtain a homotopy betweenfk−1

andfk−1 where the last map is adjoint toψk−1 ◦ φk−1 ◦ Λ(τk). It follows that there are homotopyequivalences

ξ : Xk ' Efk−1

Ξ : (XG)k ' EFk−1

making the diagram

Xk//

ξ

��

pk

##GGGGGGGGG (XG)k

Ξ

��

Pk

yysssssssss

Xk−1// (XG)k−1

Efk−1//

;;wwwwwwwwwEFk−1

eeJJJJJJJJJ

commuative. We now defineΨk andψk as the compositions

Ψk : Ek−1 ⊗τk ΛVkΨk−1⊗id−→ T ((XG)k−1)⊗Ψk−1◦τk ΛVk

ΓFk−1→ T (EFk−1)T (Ξ)−→ T ((XG)k)

ψk : Mk−1 ⊗φk−1◦τk ΛVkψk−1⊗id−→ .T (Xk−1)⊗ψk−1◦φk−1◦τk ΛVk

Γfk−1−→ T (Efk−1)T (ξ)→ T (Xk)

By the tame Hirsch lemma and Lemma 6.3, these maps arel-primary weak equivalences satisfyingconditionk+, since a primary weak equivalence satisfying condition(k − 1)+ is also a primaryweak equivalence satisfying conditionk+ and(XG)k−1 is connected by assumption. This finishesthe construction ofΨk andψk.

We now formulate the main result of this section.

Theorem 5.6. LetX be anl-tameG-space and letk ≥ 1 be a number withk < l. Then thereare (unfiltered) commutative graded differential algebras(E∗, dE) and(M∗, dM) overFl with thefollowing properties:

1) as graded algebrasE∗ = Fl[t1, . . . , tr]⊗ Λ(s1, . . . , sr)⊗M∗ where eachti is of degree2 andeachsi is of degree1.

2) the differentialdE is zero onFl[t1, . . . , tr]⊗Λ(s1, . . . , sr)⊗1 and the mapE∗ →M∗, ti, si 7→ 0,is a cochain map.

20 BERNHARD HANKE

3) M∗ is free as a graded algebra with generators in degrees1, . . . , k. As generators in degreei ∈ {1, . . . , k} we can take the elements of a basis of the freeZ(l)-moduleπi(X).

4) the cohomology algebra ofE∗ is multiplicatively isomorphic toH∗(XG;Fl) up to degreek.5) assume thatH∗(XG;Fl) vanishes in degreesk andk + 1. Then each monomial int1, . . . , tr of

cohomological degree at leastk represents the zero class inH∗(E).

Proof. After k steps the above process leads to a commutative diagram

Ekφk //

Ψk��

Mk

ψk��

T ((XG)k) // T (Xk)

where the vertical arrows arel-primary weak equivalences satisfying conditionk+. We can restrictthis diagram to filtration levelk to obtain a commutative diagram ofFl-cochain complexes

E∗,kk ⊗ Fl //

��

M∗,kk ⊗ Fl

��

T ∗,k((XG)k)⊗ Fl // T ∗,k(Xk)⊗ Fl .

The vertical arrows in this diagram arek-equivalences. By its inductive construction the cochaincomplexE∗,kk can be written as

E∗,kk =(((ΛV0 ⊗τ1 ΛV1)⊗ . . .)⊗τk ΛVk

)∗,kwhereV0 is an elementary complex of type(1, 1) andVt is an elementary complex of type(t, t)which is concentrated in degreet for t ∈ {1, . . . , k}.

We setW0 := Λ(V ∗,10 ⊗ Fl) , Wt := Λ(V ∗,tt ⊗ Fl) for 1 ≤ t ≤ k ,

regarded as unfilteredFl-cochain algebras. We hence get an unfilteredFl-cochain algebra

E∗ = ((W0 ⊗τ1 W1)⊗ . . .)⊗τk Wk .

Note thatW0 = Fl[t1, . . . , tr] ⊗ Λ(s1, . . . , sr) with trivial differential. We denote the evaluationti, si 7→ 0 by φ and obtain anFl-cochain algebra

M∗ = (W1 ⊗φ◦τ2 W2 ⊗ . . .)⊗φ◦τk Wk .

It follows thatM∗ is of the form described in Theorem 5.6, that

E∗ = Fl[t1, . . . , tr]⊗ Λ(s1, . . . , sr)⊗M∗

as graded algebras and thatE∗ →M∗, ti, si 7→ 0, is a cochain map.It remains to study the relation betweenH∗(E) andH∗(XG;Fl) as stated in parts 4) and 5) of

Theorem 5.6. Here we have to keep in mind that the product

(Ep1,q1k ⊗ Fl)⊗ (Ep2,q2

k ⊗ Fl)→ Ep1+p2,q1+q2k ⊗ Fl

is the zero map, ifq1 + q2 ≥ l. Therefore we have to be somewhat careful when comparing themultiplications in the unfiltered algebraE∗ with those in the filtered algebraE∗k .


Let B0 ⊂ V ∗,10 andBt ⊂ V t,tt be homogenousZ(l)-module bases for1 ≤ t ≤ k. We consider

the setS of monomials of cohomological degree at mostk in elements ofB0 ∪ . . . ∪ Bk. The ad-missibility of the filtration function implies thatS ⊂ E∗,kk . LetZ∗ ⊂ E∗,kk be theZ(l)-subcomplexwhich is generated by the monomials inS and their coboundaries.

We obtain inclusionsZ∗ ⊗ Fl → E∗ andZ∗ ⊗ Fl → E∗,kk ⊗ Fl of cochain complexes both ofwhich arek-equivalences. This leads to an additive isomorphism

α : H≤k(E∗) ∼= H≤k(E∗,kk ⊗ Fl) .

Consider the composition

β : H∗(E∗,kk ⊗ Fl)→ H∗(T ∗,k((XG)k)⊗ Fl)∼=→ H∗((XG)k;Fl)→ H∗(XG;Fl)

where the second map is the Cenkl-Porter map and the last map is induced by the canonical mapXG → (XG)k to thek-th Postnikov piece. As a composition ofk-equivalences the mapβ is itselfak-equivalenc.

In summary, we obtain an additive isomorphism

β ◦ α : H≤k(E∗) ∼= H≤k(E∗,kk ⊗ Fl) ∼= H≤k(XG;Fl) .

In order to study the multiplicative behaviour ofβ ◦ α, let c1 ∈ Hp1(E) andc2 ∈ Hp2(E), where

1 ≤ p1, p2 ≤ k andp1 + p2 ≤ k. Then, by the construction ofE∗,kk , the elementsα(c1) andα(c2)

can be represented by cocyclesγ1 ∈ Ep1,p1

k ⊗ Fl andγ2 ∈ Ep2,p2

k ⊗ Fl and henceα(c1 · c2) can be

represented by the cocycleγ1 · γ2 ∈ Ep1+p2,p1+p2

k ⊗ Fl ⊂ Ep1+p2,p1+p2

k ⊗ Fl ⊂ Ep1+p2,kk ⊗ Fl. Here

we have used the assumptionp1 + p2 ≤ k and the properties of our filtration function. The proofof assertion 4) is hence complete, because the diagram

Hp1(E∗,p1

k )⊗Hp2(E∗,p2

k )mult. //

β⊗β��

Hp1+p2(E∗,p1+p2

k )

β

��Hp1(XG;Fl)⊗Hp2(XG;Fl)

∪ // Hp1+p2(XG;Fl)

commutes.Now assume thatH∗(XG;Fl) vanishes in degreesk andk + 1. Let µ ∈ E∗ be a monomial in

t1, . . . , tr of cohomological degree at leastk. Because the elementst1, . . . , tr are cocycles inE2

and because the differential acts as a derivation, it is enough to assume thatµ has cohomologicaldegreek or k + 1. If the cohomological degree isk, thenµ represents the zero class inH∗(E),becauseHk(E∗) ∼= Hk(XG;Fl) = 0. If the cohomological degree isk + 1, then we argue asfollows. The elementst1, . . . , tr ∈ E2 correspond to cocyclest1, . . . , tr ∈ V 2,1

0 ⊗ Fl. By thechoice of our filtration function, we obtain a monomialµ ∈ Ek+1,k ⊗ Fl corresponding toµ.Becauseβ : Hk+1(E∗,kk ⊗Fl)→ Hk+1(XG;Fl) = 0 is injective, there is an elementν ∈ Ek,k

k ⊗Flwith µ = d(ν), whered is the differential onE∗,kk ⊗ Fl. But ν ∈ Z∗ ⊗ Fl by construction ofZ∗.BecauseZ∗ ⊗ Fl → E∗ is a cochain map, we conclude thatµ is a cocycle inE∗, which finishesthe proof of claim 5). �

If instead we start with aG-spaceX which is notl-tame, then Theorem 5.6 can still be applied inmany cases. To illustrate this letX be a connected nilpotent Kan complex with abelian fundamentalgroup and letG act onX. We denote byX(l) the l-localization ofX, see [7]. By the assumptions

22 BERNHARD HANKE

on X we haveπt(X(l)) = πt(X) ⊗ Z(l) for t ≥ 1. Furthermore, by the functoriality of thel-localization,X(l) is again aG-space and the canonicalG-equivariant mapX → X(l) induces anisomorphism

H∗((X(l))G;Fl)→ H∗(XG;Fl)

by a spectral sequence argument. Hence, ifX(l) is l-tame, then the preceding discussion applies to(X(l))G and for anyk ≥ 1 with k < l we obtain a commutative graded differential algebraE overFl which calculates theFl-cohomology ofXG up to degreek.

We specialize this discussion to theG-spaceX = Sn1 × . . .× Snk appearing in Section 2. Letl > 3 dimX. For eacht ≥ 1 with t < l this assumption implies

πt(Snj)⊗ Z(l) =

Z(l) for j > ke and t = nj

Z(l) for j ≤ ke and t = nj or t = 2nj − 1

0 otherwise

for all j ∈ {1, . . . , k}. Henceπt(X(l)) is a finitely generated freeZ(l) module of rank at mostk.Becausel > 3 dimX, we havel − 1 > k. This implies thatG acts trivially onπt(X(l)), becausethe rational representation theory [27] of the groupZ/l tells us that the smallest dimension of anontrivial rationalZ/l-representation isl − 1. We conclude thatX(l) is anl-tameG-space in thesense of Definition 5.1.

Theorem 5.6 and the subsequent discussion yield anFl-cochain algebraE∗ whose cohomologyis isomorphic toH∗(XG;Fl) in degrees less than or equal todimX + 1 and with the describedbehaviour of monomials int1, . . . , tr. This usesl > 3 dimX = dimX + 1 so that we can workwith k := dimX + 1.

It follows from Theorem 5.6 and the partial calculation ofπ∗((Snj)(l)) displayed above thatE∗

has the form described in Proposition 2.1.

6. APPENDIX: PROOF OF THE TAMEHIRSCH LEMMA

The material in this appendix is based on Sections II.7 and II.8 of the diploma thesis of Sorensen[28]. Because the proof of the tame Hirsch lemma, a result which is fundamental for our approach,is quite delicate, we provide the following detailed exposition.

Theorem 4.6 is first proven under the additional assumption that the unit

ΨΛV : ΛV → T (‖ΛV ‖)

is a primary weak equivalence satisfying conditionk+. We will show in Proposition 6.6 below thatthere is no loss of generality in assuming this.

We work in several steps. At first, we consider the untwisted case, i.e. whenf is a constant map.ThenEf = X×‖ΛV ‖ andΓf is induced by the projectionsX×‖ΛV ‖ → X,X×‖ΛV ‖ → ‖ΛV ‖,the unitΨΛV and the product of forms inT (X × ‖ΛV ‖).


Now consider the commutative diagram of filtered graded differential algebras

(6.1) T (X)⊗ ΛV

��

id⊗ΨΛV // T (X)⊗ T (‖ΛV ‖)

��

∧ // T (X × ‖ΛV ‖)

ρ

��

T (X)⊗ΛVid ⊗ΨΛV // T (X)⊗T (‖ΛV ‖)

ρ

��((T ⊗C)(X)

)⊗((T ⊗C)(‖ΛV ‖)

) ∧⊗× // (T ⊗C)(X × ‖ΛV ‖)

C(X)⊗C(‖ΛV ‖)

σ

OO

× // C(X × ‖ΛV ‖)

σ

OO

where the mapsρ andσ are taken from the zig-zag sequence (3.6) and the bar indicates the filtra-tionwise tensor product.

Lemma 6.2. The canonical map

T (X)⊗ ΛV → T (X)⊗ΛV


Proof. Forn ≥ 0 let (Λ(n)V ) ⊂ ΛV be the gradedQ-submodule generated by monomials consist-ing of exactlyn factors inV . Note thatΛ(n)V inherits a filtration fromΛV and that(Λ(n)V )∗,q = 0

for q < nk and(Λ(n)V )∗,q = (Λ(n)V )∗,nk ⊗ Qq for q ≥ nk. Furthermore, the differential onΛVrestricts to a differential onΛ(n)V .

For q ≥ 1 we have direct sum decompositions

(T (X)⊗ (ΛV ))∗,q =⊕

n≥0 , nk≤q

T ∗,q−nk(X)⊗ (Λ(n)V )∗,nk ⊗Qq

(T (X)⊗ (ΛV ))∗,q =⊕

n≥0 , nk≤q

T ∗,q(X)⊗ (Λ(n)V )∗,q .

It is therefore enough to investigate the canonical map

T ∗,q−nk(X)⊗ (Λ(n)V )∗,nk ⊗ Fl → T ∗,q(X)⊗ (Λ(n)V )∗,q ⊗ Flfor eachn ≥ 0 with nk ≤ q and any primel > q. By the algebraic Kunneth formula for chaincomplexes over the fieldFl, the induced map in cohomology splits in degreet ≥ 0 into a directsum of maps

ω : H i(T ∗,q−nk(X)⊗Fl

)⊗Hj

((Λ(n)V )∗,nk ⊗Fl

)→ H i

(T ∗,q(X)⊗Fl

)⊗Hj

((Λ(n)V )∗,q ⊗Fl

),

wherei, j ≥ 0 andi + j = t. We haveHj((Λ(n)V )∗,nk ⊗ Fl) = Hj((Λ(n)V )∗,q ⊗ Fl), becausel > q ≥ nk. Now, if q > nk, then the mapH∗(T ∗,q−nk(X) ⊗ Fl) → H∗(T ∗,q(X) ⊗ Fl) is anisomorphism by the addendum to the Cenkl-Porter Theorem 3.5 and henceω is an isomorphism.

We study the remaining caseq = nk. BecauseH∗((Λ(n)V )∗,nk ⊗ Fl) andH∗((Λ(n)V )∗,q ⊗ Fl)are concentrated in degrees at leastnd, the mapω is an isomorphism ifi + j ≤ nd and injective,if i+ j ≤ nd+ 1. Recall that fori = 1 andq = nk, the source ofω is the zero module.

First lett ≥ 1 andq ≥ t. This impliesn ≥ 1.

24 BERNHARD HANKE

If n = 1, we gett ≤ nd, because our assumptiont ≤ q = 1 · k impliest ≤ k and hencet ≤ d(recall thatd ≥ k by assumption).

If n ≥ 2, then we havet + 1 ≤ nd. For if we assumet ≥ nd, then the obvious estimatend+((t+1)−nd) ≤ t+1 and the admissibility of the filtration function implynd+(t+ 1)− nd ≤t and hencend < t. If d = 1, then this requires the assumptionn ≥ 2. But this contradicts theassumed estimatet ≤ q = nk ≤ nd.

These considerations show thatω is at-equivalence, ift ≥ 1 andq ≥ t.Now observe, that ifq ≥ k + 1, then the assumptionq = nk automatically impliesn ≥ 2.

Hence, the preceding estimate for the casen ≥ 2 shows that under this assumption,ω is actually a(t+ 1)-equivalence. This completes the proof of Lemma 6.2. �

To continue the proof of Theorem 4.6 for constantf we observe the following points. Firstly,the mapid ⊗ΨΛV : T (X)⊗ΛV → T (X)⊗T (‖ΛV ‖) is a primary weak equivalence satisfyingconditionk+. This follows from the Kunneth formula [29, Lemma 5.3.1] and the fact that byassumptionΨΛV is a primary weak equivalence satisfying conditionk+. Secondly, the universalcoefficient theorem [29, Theorem 5.3.14] implies that for eachq ≥ 1 and each primel > q, themapsρ andσ induce isomorphisms in cohomology after tensoring the cochain complexes withFl.Thirdly, the cross product map

C∗(X;Fl)⊗ C∗(‖ΛV ‖;Fl)→ C∗(X × ‖ΛV ‖;Fl)

induces an isomorphism of cohomology groups for any primel. This uses the Kunneth formula forcohomology [29, Theorem 5.6.1] and the fact that for any primel, the homologyH∗(‖ΛV ‖;Fl)is finitely generated in each degree. Recall that‖ΛV ‖ is an Eilenberg-MacLane complex for afinitely generatedQk-module, see Proposition 4.4. This completes the proof of Theorem 4.6 forconstantf .

The next step of the proof of Theorem 4.6 deals with the case whenf is homotopic to a constantmap. But before we give a proof of this case, we will show the second main technical result of thissection besides Lemma 6.2.

Lemma 6.3. LetA andB be filtered CGDAs and letV be an elementary complex of type(d, k)with d ≥ k ≥ 1. Assume thatτ : V → A is a filtered cochain map of degree1 and letf : A→ Bbe a primary weak equivalence satisfying conditionk+ so thatf : A∗,0 → B∗,0 is an isomorphism.Then the induced map

A⊗τ ΛV → B ⊗f◦τ ΛV


Proof. For q ≥ 1 let us consider the decreasing filtrationsF of (A⊗τ ΛV )∗,q and(B ⊗f◦τ ΛV )∗,q

defined forγ ≥ 0 by

Fγ(A⊗τ ΛV )∗,q = (A≥γ ⊗τ ΛV )∗,q

Fγ(B ⊗f◦τ ΛV )∗,q = (B≥γ ⊗f◦τ ΛV )∗,q .

Let l > q be a prime. The direct sum decompositionΛV =⊕

Λ(n)V from the proof of Lemma6.2 leads to the following direct sum decompositions of theE1-terms of the resulting spectral


sequences withFl-coefficients:

Ei,j1 (A) =

⊕n≥0 , nk≤q

(Ai,q−nk ⊗ Fl)⊗Hj((Λ(n)V )∗,nk ⊗ Fl)

Ei,j1 (B) =

⊕n≥0 , nk≤q

(Bi,q−nk ⊗ Fl)⊗Hj((Λ(n)V )∗,nk ⊗ Fl) .

On theE2-terms we hence get the direct sum decompositions

Ei,j2 (A) =

⊕n≥0 , nk≤q

H i(A∗,q−nk ⊗ Fl)⊗Hj((Λ(n)V )∗,nk ⊗ Fl)

Ei,j2 (B) =

⊕n≥0 , nk≤q

H i(B∗,q−nk ⊗ Fl)⊗Hj((Λ(n)V )∗,nk ⊗ Fl) ,

but notice that the differentialsd2 on theE2-terms do not in general restrict to the single summands(with varyingi andj) unlessτ is zero.

We will now prove that the map of theE2-terms induced byf is a t-equivalence, ift ≥ 1 andq ≥ t and a(t+ 1)-equivalence, ift ≥ k + 1 andq ≥ t or if t = k andq ≥ k + 1, wheret = i+ jis the cohomological degree. This implies the statement of Proposition 6.3 by induction over thepages in the spectral sequence.

First notice thatH∗((Λ(n)V )∗,nk ⊗ Fl) is concentrated in degrees at leastnd so that it suffices toshow that

ω : A∗,q−nk ⊗ Fl → B∗,q−nk ⊗ Flis a(t−nd)-equivalence, respectively a

((t+1)−nd

)-equivalence in the relevant cases. Since the

mapω is itself an isomorphism forq−nk = 0 and the mapω is a1-equivalence forq−nk ≥ 1 = 1,the mapω is a

((t + 1) − nd

)-equivalence, if−1 ≤

((t + 1) − nd

)≤ 1. Hence we will assume

from now on thatt− nd ≥ 1. Because the casen = 0 is trivial, we will also assume thatn ≥ 1.We start by analyzing the caset ≥ 1 andq ≥ t.We will show at first thatω is a (t − nd)-equivalence. Becauset − nd ≥ 1 we can use the

admissibility of the filtration function and the inequality1 + nd + (t − nd) ≤ t + 1 to conclude1 + nd + t− nd ≤ t which impliesq − nk ≥ t− nd (rememberk ≤ d). Henceω is a(t − nd)-equivalence by the assumption onf .

If in additionn ≥ 2 or if n = 1 andd > 1, then the inequalitynd+((t+1)−nd) ≤ t+1 and theadmissibility of the filtration function implyq−nk ≥ (t+ 1)− nd andω is even a

((t+1)−nd

)-

equivalence.Now let n = 1 andd = 1. This impliesk = 1. We will show thatω is still a

((t + 1) − nd

)-

equivalence, ift ≥ k + 1, q ≥ t, or if t = k, q ≥ t+ 1.Let us first concentrate on the caset ≥ k + 1 andq ≥ t. If t ≥ k + 2, thenq − nk = q − 1 ≥

t− 1 ≥ t− 1 andt− 1 ≥ k + 1, so thatω is a((t− 1) + 1

)-equivalence by the assumption onf .

In other words,ω is a((t + 1)− nd

)-equivalence as desired. Ift = k + 1, then (asq ≥ t) we get

q − nk ≥ k + 1− 1 = k + 1 (rememberk = 1), so thatω is a(k + 1)-equivalence by assumptiononf and henceω is a

((t+ 1)− nd

)-equivalence.

In the caset = k, q ≥ t + 1, we getq − nk = q − 1 ≥ t so thatω is a t-equivalence by theassumption onf and henceω is again a

((t+ 1)− nd

)-equivalence.

This finishes the proof of Lemma 6.3. �

26 BERNHARD HANKE

In order to proceed with the proof of Theorem 4.6, we assumef ' const and choose a simplicialmapH : X×I → ‖Λ(ΣV )‖ withH ◦ i0 = f andH ◦ i1 = const, whereI is the simplicial intervalandi0, i1 : X → X×I are the canonical inclusions. Forε = 0 andε = 1, we obtain a commutativediagram

EεIε //

��

EH //

��

‖Λ(coneV )‖

��

Xiε // X × I H // ‖Λ(ΣV )‖

where the horizontal maps of the first square arestrong equivalences. By definition this means thatafter applying the Cenkl-Porter functorT , one gets isomorphisms in cohomology in each degreeand in each filtration level. Using the naturality of the construction of the mapΓ, we hence get aninduced commutative diagram

T (X)⊗ Λ(V )Γconst // T (Econst)

T (X × I)⊗H] Λ(V )ΓH //

T (i0)⊗id��

T (i1)⊗id

OO

T (EH)

T (I0)

��

T (I1)

OO

T (X)⊗f] Λ(V )Γf // T (Ef )

For anyq ≥ 1 and l > q the vertical maps in this diagram restricted to filtration levelq induceisomorphisms of cohomology groups withFl-coefficients. This is clear for the right hand column.For the left hand column one argues with the spectral sequences introduced at the beginning of theproof of Lemma 6.3.

Hence, becauseΓconst is a primary weak equivalence satisfying conditionk+, the same is truefor the mapΓf .

Now these special cases are used to show the tame Hirsch lemma for arbitrary simplicial setsXby a Mayer-Vietoris argument and induction over them-skeleta ofX.

We consider simplicial mapsf : X → ‖Λ(ΣV )‖ whereX is a simplicial set andV is anelementary complex of type(d, k) with d ≥ k ≥ 1. In the following we leaveV fixed, but considervaryingX.

First, the tame Hirsch lemma holds, ifX is a disjoint union of simplicialm-simplices,m ≥ 0.This follows from the preceding line of arguments, because in this casef is homotopic to a constantmap. Recall that‖Λ(ΣV )‖ is connected and fulfills the Kan condition by Proposition 4.4. Inparticular, the tame Hirsch lemma holds, ifX is a0-dimensional simplicial set. We now assumethat the tame Hirsch lemma has been proven for any(m− 1)-dimensional simplicial setX, wherem ≥ 1. If X ism-dimensional, we writeX as a push out∐

∂∆[m] //

��

∐∆[m]

��Xm−1 // X

and do the same forEf . Here∂∆[m] denotes the simplicial(m − 1)-sphere. Forq ≥ 1 weconsider the following commutative diagram, where the columns are Mayer-Vietoris sequences


and the horizontal arrows are induced byΓf or appropriate restrictions thereof:

(6.4) 0

��

0

��(T (X)⊗f] ΛV

)∗,q��

// T ∗,q(Ef )

��(T (Xm−1)⊗f] ΛV

)∗,q ⊕ (T (∐

∆[m])⊗f] ΛV)∗,q

��

// T ∗,q(Ef |Xm−1)⊕ T ∗,q(Ef |∐∆[m])

��(T (∐∂∆[m])⊗f] ΛV

)∗,q��

// T ∗,q(Ef |∐ ∂∆[m])

��0 0

.

The right-hand column is exact becauseT ∗,q is a cohomology theory forq ≥ 1. Unfortunately,this is not quite true for the left-hand column: indeed, for any simplicial setY , we have a directsum decomposition of gradedQq-modules

(6.5) (T (Y )⊗ (ΛV ))∗,q =⊕

n≥0 , nk≤q

T ∗,q−nk(Y )⊗ (Λ(n)V )∗,nk ⊗Qq

as in the proof of Lemma 6.2. Furthermore, ifq − nk ≥ 1 we have a short exact Mayer-Vietorissequence

0→ T ∗,q−nk(X)→ T ∗,q−nk(Xm−1)⊕ T ∗,q−nk(∐

∆[m])→ T ∗,q−nk

(∐∂∆[m]

)→ 0 ,

which stays exact after tensoring with(Λ(n)V )∗,nk. However, forq = nk we need to be careful:the sequence

0→ T ∗,0(X)→ T ∗,0(Xm−1)⊕ T ∗,0(∐

∆[m])→ T ∗,0

(∐∂∆[m]

)is indeed exact for allm ≥ 1, but the final map is surjective, only ifm ≥ 2, in general. For anyq′ ≥ 0 let us therefore consider the cochain complex

T∗,q′(∐

∂∆[m])

:= im((T ∗,q

′(Xm−1)⊕ T ∗,q′(

∐∆[m])

)→ T ∗,q

′(∐

∂∆[m])),

which coincides withT ∗,q′(∐∂∆[m]) if q′ ≥ 1 or if m ≥ 2. For any primel > q we get an

inclusion of cochain complexes(T(∐

∂∆[m])⊗f] ΛV

)∗,q⊗ Fl ↪→

(T(∐

∂∆[m])⊗f] ΛV

)∗,q⊗ Fl ,

which is a(t − 1)-equivalence fort ≥ 1 andq ≥ t and is at-equivalence fort ≥ k andq ≥max{t, k + 1}. This is implied by the direct sum decomposition (6.5) and an argument similarto the one for the caseq = nk in the proof of Lemma 6.2. Furthermore, the left hand column inDiagram (6.4) becomes exact after replacingT (

∐∂∆[m])⊗f] ΛV by T (

∐∂∆[m])⊗f] ΛV .

After these preliminaries a five lemma argument together with the induction hypothesis provesthatΓf : T (X) ⊗f] Λ(V ) → T (Ef ) is a primary weak equivalence satisfying conditionk+, thusfinishing the induction step.

28 BERNHARD HANKE

Finally, we assume thatX is not of finite dimension. In this case we observe that the restrictionmaps

(T ∗,q(X)⊗f] ΛV

)⊗Fl →

(T ∗,q(Xm)⊗f] ΛV

)⊗Fl andT ∗,q(Ef )⊗Fl → T ∗,q(Ef |Xm)⊗Fl

induce isomorphisms in cohomology up to degreem− 1 for any primel > q.As a first application of the tame Hirsch lemma we study Eilenberg-MacLane complexes. The

following result says that the assumption on the unitΨΛV upon which the above proof of the tameHirsch lemma was based always holds.

Proposition 6.6. LetV be an elementary complex of type(d, k) with d ≥ k ≥ 1. Then the unit

ΨΛV : ΛV → T (‖ΛV ‖)


The proof starts with the following lemma, which formulates one of the guiding principles oftame homotopy theory.

Lemma 6.7. Letd, q ≥ 1 and letl be a prime withl > q. LetΛ(v) be the free (unfiltered) gradedcommutativeFl-algebra generated by a variablev of degreed. Then the algebra map

Λ(v)→ H∗(K(Qq, d);Fl)

induced by sendingv to the canonical generator ofHd(K(Qq, d);Fl) is an isomorphism in degreeless than or equal tod+ 2l − 3 and is an isomorphism in all degrees, ifd = 1.

Proof. The result is clear, ifd = 1, because then both sides are concentrated in degree1. Foran arbitrary primel, a classical result of Cartan and Serre says thatH∗(K(Z, d);Fl) is a freecommutative graded algebra overFl in one generator of degreed and further generators of degreesat leastd+ 2(l − 1) as the first reduced Steenrod power operation for the primel raises degree by2(l − 1).

The inclusionZ ↪→ Qq induces an isomorphismH∗(K(Qq, d);Fl)→ H∗(K(Z, d);Fl), becausel > q. This finishes the proof of Lemma 6.7. �

Proposition 6.8. Proposition 6.6 holds ifV is concentrated in degreed andV d,k ∼= Qk.

Proof. We start by noting that for eachq′ ≥ k and for any primel > q′, the unitΨΛV induces anisomorphism

(6.9) Hd((ΛV )∗,q′ ⊗ Fl)→ Hd(T ∗,q

′(‖ΛV ‖)⊗ Fl)

by Proposition 4.5 and Theorem 3.5.We first treat the cased = 1 separately. In this case we havek = 1. Let q ≥ 1 and l > q.

Thenq ≥ k and the cohomologyH∗(Λ(V )∗,q ⊗ Fl) is concentrated in degree1 and isomorphic toFl. The same holds forH∗(T ∗,q(‖Λ(V )‖) ⊗ Fl) by the Cenkl-Porter theorem. Hence,ΨΛV is aprimary weak equivalence satisfying conditionk+.

From now on, we assumed ≥ 2. Let t ≥ 1 andq ≥ t. We distinguish the casest < k andt ≥ k. In the first case we havet < d (recall k ≤ d) and we will show thatΨΛV : (ΛV )∗,q ⊗Fl → T ∗,q(‖V ‖) ⊗ Fl is a t-equivalence forl > q. That the induced map in cohomology is anisomorphism up to degreet is trivial, becauset < d. The injectivity in degreet + 1 is againtrivial, if q < k, because(ΛV )t+1,q = 0 in this case. Forq ≥ k this injectivity follows from theisomorphism (6.9).

It remains to study the caset ≥ k. Let l > q be a prime. We will show thatΨΛV : (ΛV )∗,q⊗Fl →T ∗,q(‖V ‖)⊗ Fl is a(t+ 1)-equivalence. This will finish the proof of Proposition 6.8.


It follows from the Cenkl-Porter theorem and from Lemma 6.7 that for allk ≤ q′ ≤ q the gradedgroupsH∗(T ∗,q

′(‖Λ(V )‖)⊗Fl) andΛ(v)⊗Fl are canonically isomorphic up to degreed+ 2l− 3,

wherev denotes a variable of degreed. Furthermore, these isomorphisms are compatible withmultiplications on filtration levelsq′1 andq′2 with k ≤ q′1, q

′2 ≤ q andq′1 + q′2 ≤ q. Note that the

assumptionst ≥ k andl > q ≥ t imply l > k so that Lemma 6.7 can be applied. Asd ≥ 2 andl ≥ 2, we obtain the inequalityd + 2l − 3 ≥ l + 1. And becausel > q ≥ t ≥ t, the isomorphismH∗(T ∗,q

′(‖Λ(V )‖)⊗Fl) ∼= Λ(v)⊗Fl holds up to degreet+2. For a proof of the assertion thatΨΛV

is a(t+ 1)-equivalence, it remains to show that for alln ≥ 1 with nd ≤ t+ 1 we get the inequalitynk ≤ q. For then we can apply induction onn starting with the isomorphism (6.9) withq′ = k.Notice that in degreet+2, we need not care about the casenk > q, because then(ΛV )t+2,q⊗Fl = 0so that the map in question is certainly injective. Forn = 1 the required inequality holds, becausek ≤ t ≤ q. Forn ≥ 2 it follows from the chain of inequalitesnk ≤ nd ≤ t ≤ q where the secondinequality uses the admissibility of the filtration function and the assumptionsd, n ≥ 2.

This finishes the proof of Proposition 6.8. �

We can now assemble the proof of Proposition 6.6.First, letV be concentrated in degreed. We proceed by induction on the rank ofV d,k. If this

rank is equal to1, we apply Proposition 6.8. Now letV d,k be a freeQk-module of rankn+ 1 withgeneratorsv1, . . . , vn+1. Applying the tame Hirsch lemma and Proposition 6.8, the map

T (‖Λ(v1, . . . , vn)‖)⊗ Λ(vn+1)→ T (‖Λ(v1, . . . , vn)‖ × ‖Λ(vn+1)‖) = T (‖ΛV ‖)

is a primary weak equivalence satisfying conditionk+. HereΛ(vn+1) is the free filtered CGDAgenerated byvn+1. Hence we need to show that the map

Λ(v1, . . . , vn)⊗ Λ(vn+1)→ T (‖Λ(v1, . . . , vn)‖)⊗ Λ(vn+1)

is a primary weak equivalence satisfying conditionk+. But this follows from Lemma 6.3 andthe induction hypothesis. The assumption on filtration level0 holds, because‖Λ(v1, . . . , vn)‖ isconnected.

We are now able to prove the general case of Proposition 6.6. Let(V, η) be an elementarycomplex of type(d, k) with d ≥ k ≥ 1. We can splitV into two elementary complexesV0 oftype(d, k) andV1 of type(d + 1, k) which are concentrated in degreesd andd + 1, respectively.The differentialη in V is then regarded as a map of filtered cochain complexesη : ΣV0 → V1.Applying the simplicial realization functor yields a simplicial map

‖Λ(η)‖ : ‖ΛV1‖ → ‖Λ(ΣV0)‖ .

Because the unitΨΛV0 is a primary weak equivalence satisfying conditionk+, which has alreadybeen shown, the tame Hirsch lemma applies and we get a primary weak equivalence satisfyingconditionk+

Γ‖Λ(η)‖ : T (‖ΛV1‖)⊗ΨΛV1◦η ΛV0 → T (E‖Λ(η)‖) .

By an argument similar to the one used in the proof of Proposition 4.4, we see thatE‖Λ(η)‖ = ‖ΛV ‖and by Lemma 6.3, the map

ΛV = ΛV1 ⊗η ΛV0 → T (‖ΛV1‖)⊗ΨΛV1◦ηΛV0

is a primary weak equivalence satisfying conditionk+. This finishes the proof of Proposition 6.6.

30 BERNHARD HANKE

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UNIVERSITAT M UNCHEN, GERMANY ,E-mail address: [email protected]

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