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arXiv:1105.3951v1 [math.AT] 19 May 2011

A SIMPLE HOMOTOPY-THEORETICAL PROOF OF THE

SULLIVAN CONJECTURE

JEFFREY STROM

Abstract. We give a new proof, using comparatively simple tech-

niques, of the Sullivan conjecture: map∗(BZ/p,K) ∼ ∗ for every finite-

dimensional CW complex K.

Introduction

Haynes Miller proved the Sullivan conjecture in [15]. Another proof can

be deduced from the extension due to Lannes [10], whose proof depends on

the insights into unstable modules and algebras afforded by the T functor.

There are three main problems with these proofs: they are very complicated;

the content is almost entirely encoded in pure algebra; and it is difficult to

tease out the fundamental properties of BZ/p that make the proofs work.

Our purpose in this paper is to offer a new proof avoiding these com-

plaints. The Sullivan conjecture is an easy consequence of the following

main theorem.

Theorem 1. Let X be a CW complex of finite type, and assume that

? H∗(X;Z[1

(1)? H∗(X;Fp) is a reduced unstable Ap-module and? H∗(X;Fp)⊗J(n) is

an injective unstable Ap-module for all n ≥ 0.

(2) Exts

(3) map∗(X,S2m+1) ∼ ∗ for all sufficiently large m.

(4) map∗(X,K) ∼ ∗ for all simply-connected finite complexes K.

(5) map∗(X,?∞

(6) map∗(X,K) ∼ ∗ for all simply-connected finite-dimensional CW

complexes K.

(7) map∗(X,K) ∼ ∗ for all simply-connected CW complexes K with

clW(K) < ∞.

Furthermore, if π1(X) has no perfect quotient groups (that is, if π1(X) is

hypoabelian), then the simply-connected hypotheses on K are not needed.

p]) = 0. Then each of the following conditions implies the next.

U(Σ2m+1Fp,? H∗(Σs+tX)) = 0 for every s,t ≥ 0 and all m ≥ 0.

i=1Sni) ∼ ∗ for any countable set {ni} with each ni> 1.

The notation clW(K) in part (7) denotes the cone length of K with respect

to the collection W of all wedges of spheres; see [17, §1] for a brief overview

2010 Mathematics Subject Classification. Primary 55S37, 55R35; Secondary 55S10.

Key words and phrases. Sullivan conjecture, homotopy limit, cone length, phantom

map, Massey-Peterson tower, T functor, Steenrod algebra, unstable module.

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2 J. Strom

of the main properties of cone length. Since (7) implies (3), the last five

statements are actually equivalent.

Every finite-dimensional CW complex K of course has finite cone length

with respect to W. The extension from part (6) to part (7) is a geometric

parallel to the passage from finite-dimensional spaces in [15] to spaces with

locally finite cohomology in [10].

Since it is known that the reduced cohomology? H∗(BZ/p;Fp) is reduced

[16, Lem. 2.6.5] and that? H∗(BZ/p;Fp) ⊗ J(n) is an injective unstable Ap-

module for all n ≥ 0 [16, Thm. 3.1.1], we obtain the Sullivan conjecture as

an immediate consequence.

Corollary 2 (Miller). If clW(K) < ∞ (and, in particular, if K is finite-

dimensional), then map∗(BZ/p,K) ∼ ∗.

The claim that our proof is simple should be justified: we make no use

whatsoever of spectral sequences, except implicitly in making use of the

well-known cohomology of Eilenberg-Mac Lane spaces; the existence of un-

stable algebras over the Steenrod algebra is mentioned only to make sense of

Massey-Peterson towers; finally, the only homological algebra in this paper

is the usual abelian kind. It has seemed appropriate at points to emphasize

the simplicity of the present approach by laying out some results that might

just as well have been cited from their sources.

Acknowledgement. Many thanks are due to John Harper for pointing out

that I had the raw materials for a proof of the full Sullivan conjecture, and

for bringing the paper [14] to my attention.

1. Preliminaries

We begin by reviewing some preliminary material on the category U of

unstable Ap-algebras, Massey-Peterson towers and phantom maps.

1.1. Unstable Modules over the Steenrod Algebra. The cohomology

functor H∗(?;Fp) takes its values in the category U of unstable modules and

their homomorphisms. An unstable module over the Steenrod algebra Ap

is a graded Ap-module M satisfying PI(x) = 0 if e(I) > |x|, where e(I) is

the excess of I and |x| is the degree of x ∈ M. We begin with some basic

algebra of unstable modules, all of which is (at least implicitly) in [16].

Suspension of Modules. An unstable module M ∈ U has a suspension

ΣM ∈ U given by (ΣM)n= Mn−1. The functor Σ : U → U has a left

adjoint Ω and a right adjoint?Σ. A module M is called reduced if?ΣM = 0.

Projective and Injective Unstable Modules. In the category U, there

are free modules F(n) = Ap/E(n), where E(n) is the smallest left ideal

containing all Steenrod powers PIwith excess e(I) > n. It is easy to see

that the assignment f ?→ f([1]) defines natural isomorphisms

HomU(F(n),M)

∼=

− → Mn.

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A simple homotopy-theoretical proof of the Sullivan conjecture3

This property defines F(n) up to natural isomorphism, and shows that F(n)

deserves to be called a free module on a single generator of dimension n.

More generally, the free module on a set X = {xα} with |xα| = nαis (up to

isomorphism) the sum?F(nα) (see [16, §1.6] for details).

A graded Fp-vector space M is of finite type if dimFp(Mk) < ∞ for each

k. Since Apis of finite type, so is F(n).

The functor which takes M ∈ U and returns the dual Fp-vector space

(Mn)∗is representable: there is a module J(n) ∈ U and a natural isomor-

phism

HomU(M,J(n))− → HomFp(Mn,Fp).

Since finite sums of vector spaces are also finite products, these functors are

exact, so the module J(n) is an injective object in U.

∼=

The Functor τ. In [16, Thm 3.2.1] it is shown that for any module H ∈ U,

the functor H⊗Ap? has a left adjoint, denoted (? : H)U. Fix a module H (to

stand in for? H∗(X;Fp)) and write τ for the functor (? : H)U; this is intended

to evoke the standard notation T for the special case H =? H∗(BZ/p;Fp).

Lemma 3. Let H ∈ U be a reduced unstable module of finite type and

suppose that H ⊗ J(n) is injective in U for every n ≥ 0. Then

(a) τ exact,

(b) τ commutes with suspension,

(c) if M is free and of finite type, then so is τ(M), and

(d) if H0= 0, then τ(M) = 0 for any finite module M ∈ U.

Proof. These results are covered in Sections 3.2 and 3.3 of [16]. Specifically,

parts (a) and (b) are proved as in [16, Thm. 3.2.2 & Prop. 3.3.4]. Parts (c)

and (d) may be proved following [16, Lem. 3.3.1 & Prop. 3.3.6], but since

there are some changes needed, we prove those parts here.

Write dk= dimFp(Hk); then there are natural isomorphisms

∼=

HomU(F(n),H ⊗ M)

∼=

HomU

HomU(τ(F(n)),M)

??

i+j=nF(i)⊕dj,M

?

,

proving (c) in the case of a free module on one generator. Since τ is a left

adjoint, it commutes with colimits (and sums in particular), we derive the

full statement of (c).

If H0= 0, then d0 = 0 and τ(F(n)) is a sum of free modules F(k)

with k < n. Since F(0) = Fp, we see that τ(Fp) = 0; then (a), together

with the fact that τ commutes with colimits, implies that τ(M) = 0 for all

trivial modules M. Finally, any finite module M has filtration all of whose

subquotients are trivial, and (d) follows.

?

1.2. Massey-Peterson Towers. Cohomology of spaces has more structure

than just that of an unstable Ap-module. It has a cup product which makes

H∗(X;Fp) into an unstable algebra over Ap. The category of unstable

algebras is denoted K.

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4 J. Strom

The forgetful functor K → U has a left adjoint U : U → K. A space X

is said to have very nice cohomology if H∗(X)∼= U(M) for some unstable

module M of finite type.

Since U(F(n))∼= H∗(K(Z/p,n)), there is a contravariant functor K which

carries a free module F to a generalized Eilenberg-Mac Lane space (usually

abbreviated GEM) K(F) such that H∗(K(F))∼= U(F). If F is free, then

so is ΩF, and K(ΩF) ≃ ΩK(F).

Lemma 4. For any X, [X,K(F)]∼= HomU(F,? H∗(X)).

It is shown in [8,11,14] that if H∗(Y )∼= U(M) and P∗→ M → 0 is a free

resolution in U, then Y has a Massey-Peterson tower

···

??Ys

??

??

Ys−1

??

??

···

??Y1

??

??

Y0

??

K(ΩsPs+1)

K(Ωs−1Ps)

K(ΩP2)K(P1)

in which

(1) Y0= K(P0),

(2) each homotopy group πk(Ys) is a finite p-group,

(3) the limit of the tower is the completion Y∧

(4) each sequence Ys→ Ys−1→ K(Ωs−1Ps) is a fiber sequence, and

(5) the compositions ΩK(Ωs−1Ps) −→ Ys−→ K(ΩsPs+1) can be nat-

urally identified with K(Ωsds+1), where ds+1 : Ps+1 → Ps is the

differential in the given free resolution.

p,

1.3. Phantom Maps. A phantom map is a map f : X → Y from a CW

complex X such that the restriction f|Xnof f to the n-skeleton is trivial for

each n. We write Ph(X,Y ) ⊆ [X,Y ] for the set of pointed homotopy classes

of phantom maps from X to Y . See [12] for an excellent survey on phantom

maps.

If X is the homotopy colimit of a telescope diagram ··· → X(n)→

X(n+1)→ ···, then there is a short exact sequence of pointed sets

∗ → lim1[ΣX(n),Y ] −→ [X,Y ] −→ lim[X(n),Y ] → ∗,

and dually, if Y is the homotopy limit of a tower ··· ← Y(n)← Y(n+1)← ···,

then there is a short exact sequence

∗ → lim1[X,ΩY(n)] −→ [X,Y ] −→ lim[X,Y(n)] → ∗.

In the particular case of the expression of a CW complex X as the homotopy

colimit of its skeleta or of a space Y as the homotopy limit of its Postnikov

system, the kernels are the phantom sets.

We will be interested in showing that all phantom maps are trivial. One

useful criterion is that if G is a tower of compact Hausdorff topological

spaces, then lim1G = ∗ (see [12, Prop. 4.3]). This is used to prove the

following lemma.

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A simple homotopy-theoretical proof of the Sullivan conjecture5

Lemma 5. Let ··· ← Ys← Ys+1← ··· be a tower of spaces such that each

homotopy group πk(Ys) is finite. If Z is of finite type, then lim1[Z,ΩYs] = ∗.

Proof. The homotopy sets [Zn,ΩjYs] are finite, and we give them the discrete

topology, resulting in towers of compact groups and continuous homomor-

phisms. Fixing s and lettting n vary, we find that lim1[Zn,Ω2Ys] = ∗, and

hence the exact sequence

0 → lim1n[Zn,Ω2Ys] −→ [Z,ΩYs] −→ limn[Zn,ΩYs] → 1

(of groups) reduces to an isomorphism [Z,ΩYs]∼= limn[Zn,ΩYs].

[Z,ΩYs] is an inverse limit of finite discrete spaces, it is compact and Haus-

dorff; and since the structure maps Ys→ Ys−1induce maps of the towers

that define the topology, the induced maps [Z,ΩYs] → [Z,ΩYs−1] are con-

tinuous. Thus lim1[Z,ΩYs] = ∗.

Since

?

The Mittag-Leffler condition is another useful criterion for the vanishing

of lim1. A tower of groups ··· ← Gn ← Gn+1 ← ··· is Mittag-Leffler

if there is a function κ : N → N such that for each n Im(Gn+k→ Gn) =

Im(Gn+κ(n)→ Gn) ⊆ Gnfor every k ≥ κ(n).

Proposition 6. Let ··· ← Gn← Gn+1← ··· be a tower of groups.

(a) If the tower is Mittag-Leffler, then lim1Gn= ∗.

(b) If each Gnis a countable group, then the converse holds: if lim1Gn= ∗,

then the tower is Mittag-Leffler [12, Thm. 4.4].

Importantly, the Mittag-Leffler condition does not refer to the algebraic

structure of the groups Gn. This observation plays a key role in the following

result (cf. [13, §3]).

Proposition 7. Let X be a CW complex of finite type, and let Y1and Y2

be countable CW complexes with ΩY1≃ ΩY2. Then Ph(X,Y1) = ∗ if and

only if Ph(X,Y2) = ∗.

Proof. The homotopy equivalence ΩY1 ≃ ΩY2 gives levelwise bijections

{[ΣXn,Y1]}∼= {[Xn,ΩY1]}∼= {[Xn,ΩY2]}∼= {[ΣXn,Y2]} of towers of sets.

Since X is of finite type and Y1,Y2are countable CW complexes, these tow-

ers are towers of countable groups. Now the triviality of the first phantom

set implies that the first tower is Mittag-Leffler; but then all four towers

must be Mittag-Leffler, and the result follows.

?

2. (1) implies (2)

Write H =? H∗(X;Fp); thus H ∈ U is a reduced module of finite type and

H ⊗ J(n) is injective for all n. If P∗→ M → 0 is a free resolution of M in