Page 1

?This research was supported in part by an Alfred P. Sloan Dissertation Fellowship.

*Fax: #617-253-4995.

E-mail address: mandell@math.mit.edu (M.A. Mandell)

Topology 40 (2001) 43}94

E?algebras and p-adic homotopy theory?

Michael A. Mandell*

Department of Mathematics, Massachusetts Institute of Technology, Room 2-265, 77 Massachusetts Avenue, Cambridge,

MA 02139, USA

Received 7 February 1998; received in revised form 1 November 1998; accepted 24 April 1999

Abstract

Let F?denote the "eld with p elements and F M?its algebraic closure. We show that the singular cochain

functor with coe$cients in F M?induces a contravariant equivalence between the homotopy category of

connected p-complete nilpotent spaces of "nite p-type and a full subcategory of the homotopy category of

E?F M?-algebras. ? 2000 Published by Elsevier Science Ltd. All rights reserved.

MSC: primary 55P15; secondary 55P60

Introduction

Since the invention of localization and completion of topological spaces, it has proved extremely

useful in homotopy theory to view the homotopy category from the perspective of a single prime at

a time. The work of Serre, Quillen, Sullivan, and others showed that, viewed rationally, homotopy

theory becomes completely algebraic. In particular, Sullivan showed that an important sub-

category of the homotopy category of rational spaces is contravariantly equivalent to a sub-

category of the homotopy category of commutative di!erential graded Q-algebras, and that the

functor underlying this equivalence is closely related to the singular cochain functor. In this paper,

we o!er a similar theorem for p-adic homotopy theory.

Since the non-commutativity of the multiplication of the F?singular cochains is visible already

on the homology level in the Steenrod operations, one would not expect that any useful sub-

category of the p-adic homotopy category to be equivalent to a category of commutative

0040-9383/01/$-see front matter ? 2000 Published by Elsevier Science Ltd. All rights reserved.

PII: S 0 0 4 0 - 9 3 8 3 ( 9 9 ) 0 0 0 5 3 - 1

Page 2

di!erential graded algebras. We must instead look to a more sophisticated class of algebras, the

E?algebras [18]. E?algebras, roughly, are di!erential graded modules with an in"nitely coherent

homotopy associative and commutative multiplication. They provide a generalization of com-

mutative di!erential graded algebras that admits homology operations as commutativity obstruc-

tions generalizing the Steenrod operations. To capture p-adic homotopy theory, even the category

of E?F?-algebras is not quite su$cient; rather we consider E?algebras over the algebraic closure

F M?of F?. We prove the following theorem.

Main Theorem. The singular cochain functor with coezcients in F M?induces a contravariant equiva-

lence from the homotopy category of connected p-complete nilpotent spaces of xnite p-type to a full

subcategory of the homotopy category of E?F M?-algebras.

The homotopy category of connected p-complete nilpotent spaces of "nite p-type is a full

subcategoryof the p-adic homotopy category, the category obtainedfrom the categoryof spacesby

formally inverting those maps that induce isomorphisms on singular homology with coe$cients in

F?. The p-adic homotopy category itself can be regarded as a full subcategory of the homotopy

category, the category obtained from the category of spaces by formally inverting the weak

equivalences. We remind the reader that a connected space is p-complete, nilpotent, and of "nite

p-type if and only if its Postnikov tower has a principal re"nement in which each "ber is of type

K(Z/pZ,n) or K(Z??,n), where Z??denotes the p-adic integers.

By the homotopy category of E?F M?-algebras, we mean the category obtained from the category

of algebras over a particular but unspeci"ed E?F M?operad by formally inverting the maps in that

category that are quasi-isomorphisms of the underlying di!erential graded F M?-modules, the maps

that induce an isomorphism of homology groups. It is well-known that up to equivalence, this

category does not depend on the operad chosen. We refer the reader to [18, I] for a good

introduction to operads, E?operads, and E?algebras.

To complete the picture, we need to identify intrinsically the subcategory of the homotopy

category of E?F M?-algebras that the Main Theorem asserts an equivalence with. Although we can

write a necessary and su$cient condition for an E?F M?-algebra to be quasi-isomorphic to the

singular cochain complex of a connected p-complete nilpotent space of "nite p-type, it is relatively

unenlightening and di$cult to verify in practice. This condition is stated precisely in Section 7 and

is essentially the E?F M?-algebra analogue of the existence of a "nite p-type principal Postnikov

tower. Unsurprisingly, restricting consideration to simply connected spaces makes the identi"ca-

tion signi"cantly easier. In fact, we can write necessary and su$cient conditions for an E?F M?-

algebra to be quasi-isomorphic to the singular cochain complex of a 1-connected space of "nite

p-type in terms of its homology and the generalized Steenrod operation P?.

Characterization Theorem. An E?diwerential graded F M?-algebra A is quasi-isomorphic in the

category of E?F M?-algebras to the singular cochain complex of a 1-connected (p-complete) space of

xnite p-type if and only if H?A is zero for i(0, H?A"F M?, H?A"0, and for i'1, H?A is xnite

dimensional over F M?and generated as an F M?-module by the xxed points of the operation P?.

Succinctly, the Characterization Theorem states that an E?F M?-algebra A is quasi-isomorphic to

the singular cochain complex of a 1-connected space of "nite p-type if and only if the homology of

44

M.A. Mandell / Topology 40 (2001) 43}94

Page 3

A looks like the cohomology of such a space as a module over the generalized Steenrod

algebra.

Comparison with other approaches

The papers [13,17,26] and the unpublished ideas of Dwyer and Hopkins [8] all compare p-adic

homotopy theory to various homotopy categories of algebras (or coalgebras). We give a short

comparison of these results to the results proved here.

The "rst announced results along the lines of our Main Theorem appeared in [26]. The

arguments there are not well justi"ed, however, and some of the results appear to be wrong.

More recently, Goerss [13,17] has compared the p-adic homotopy category with the homotopy

categories of simplicial cocommutative coalgebras and cosimplicial commutative algebras. In

particular, Goerss [13] proves that the p-adic homotopy category embeds as a full subcategory of

the homotopy category of cocommutative simplicial F M?-coalgebras. The analogue of the Charac-

terization Theorem is not known in this context. It is straightforward to describe the relationship

between the results of Goerss [13] and our Main Theorem. There is a functor from the homotopy

category of simplicial cocommutative coalgebras to the homotopy category of E?algebras given

by normalization of the dual cosimplicial commutative algebra [15] (see also Section 1 below).

Applied to the singular simplicial chains of a space, we obtain the singular cochain complex of that

space. Our Main Theorem implies that on the subcategory of nilpotent spaces of "nite p-type, this

re"ned functor remains a full embedding. This gives an a$rmative answer to the question asked in

[17, 6.3].

The unpublished ideas of Dwyer and Hopkins [8] for comparing the p-adic homotopy category

to the homotopy category of E?ring spectra under the Eilenberg}MacLane spectrum HF M?, would

give a `brave new algebraa version of our Main Theorem. A proof of such a comparison can be

given along similar lines to the proof of our Main Theorem. We sketch the argument in Appendix

C. Theanalogueof the CharacterizationTheoremin this contextwas not consideredin [8],but can

be proved by essentially the same arguments as the proof of our Characterization Theorem.

A direct comparison between our approach and this approach to p-adic homotopy theory would

require a comparison of the homotopy category of E?HF M?ring spectra and the category of

E?F M?-algebras, and also an identi"cation of the composite functor from spaces to E?di!erential

graded F M?-algebras as the singular cochain functor. We will provide this comparison and this

identi"cation in [19,20].

1. Outline of the paper

Since the main objects we work with in this paper are the cochain complexes, it is convenient to

grade di!erential graded modules `cohomologicallya with the di!erential raising degrees. This

makes the cochain complexes concentrated in non-negative degrees, but forces E?operads to be

concentrated in non-positive degrees. Along with this convention, we write the homology of

a di!erential graded module M as HHM. We work almost exclusively with ground ring F M?;

throughout this paper, CHX and HHX always denote the cochain complex and the cohomology of

X taken with coe$cients in F M?. We write CH(X;F?) and HH(X;F?) for the cochain complex and the

M.A. Mandell / Topology 40 (2001) 43}94

45

Page 4

cohomology of X with coe$cients in F?or CH(X;k) and HH(X;k) for these with coe$cients in

a commutative ring k.

The "rst prerequisite to the Main Theorem is recognizing that the singular cochain functor can

be regarded as a functor into the category of E-algebras for some E?F M?-operad E. In fact, for the

purpose of this paper, the exact construction of this structure does not matter so long as the

(normalized) cochain complex of a simplicial set is naturally an E-algebra. However, we do need to

know that such a structure exists. This can be shown as follows.

The work of Hinich and Schechtman in [15] gives the singular cochain complex of a space or

the cochain complex of a simplicial set the structure of a `May algebraa, an algebra over an

acyclic operad Z, the `Eilenberg}Zilbera operad. The operad Z is not an E?operad however

since it is not ?-free and since it is non-zero in both positive and negative degrees. To "x this, let

Z M be the `(co)-connective covera of Z: Z M (n) is the di!erential graded F M?-module that is equal to

Z(n) in degrees less than zero, equal to the kernel of the di!erential in degree zero, and zero in

positive degrees. The operadic multiplication of Z lifts to Z M , making it an acyclic operad.

Tensoring Z M with an E?operad C gives an E?operad C and a map of operads EPZ. The

cochain complex of a simplicial set then obtains the natural structure of an algebra over the

E?operad E.

We write E for the category of E-algebras.Since we are assuming that the functorCH from spaces

to E-algebras factors through the category of simplicial sets, we can work simplicially. As is fairly

standard, we refer to the category obtained from the category of simplicial sets by formally

inverting the weak equivalences as the homotopy category; this category is equivalent to the

category of Kan complexes and homotopy classes of maps and to the category of CW spaces and

homotopy classes of maps. Since the cochain functor converts F?-homology isomorphisms and in

particular weak equivalences of simplicial sets to quasi-isomorphisms of E-algebras, the (total)

derived functor exists as a contravariant functor from the homotopy category to the homotopy

category of E-algebras. We prove the Main Theorem by constructing a right adjoint U from the

homotopy category of E-algebras to the homotopy category and showing that it provides an

inverse equivalence on the subcategories in question.

In order to construct the functor U and to analyze the composite UCH, we need some tools to

help us understand the homotopy category of E-algebras. The tools we need are precisely those

provided by Quillen's theory of closed model categories [25] (see also [9]). Unfortunately, we have

not been able to verify that the category of E-algebras is a model category. Nevertheless, the

category of E-algebras is close enough that most of the standard model category arguments apply,

and we obtain the results we need. These theorems are summarized in Section 2.

Various steps in the proofs of the Main Theorem and the Characterization Theorem require

understanding of the derived coproduct and the homotopy pushout of E-algebras. We summarize

the results we need in Section 3; the proofs of these results are in Section 14.

We construct in Section 4 a contravariant functor ; from the category of E-algebras to the

category of simplicial sets that is the right adjoint to CH. Our model theoretic results allow us to

show that the right derived functor of ; exists and is right adjoint to the derived functor of CH; this

derived functor is our functor U mentioned above. Precisely, U is a contravariant functor from the

homotopy category of E-algebras to the homotopy category, and we have a canonical isomorphism

H?(X,UA)?h M E(A,CHX)

46

M.A. Mandell / Topology 40 (2001) 43}94

Page 5

for a simplicial set X and an E-algebra A. Here and elsewhere, H?denotes the homotopy category

and h M E denotes the homotopy category of E-algebras.

We write u?for the `unita of the derived adjunction XPUCHX. For the purposes of this paper,

we say that a simplicial set X is resolvable by E?F M?-algebras or just resolvable if the map u?is an

isomorphism in the homotopy category. In Section 5, we prove the following two theorems.

Theorem 1.1. Let X be the limit of a tower of Kan xbrations 2PX?P2X?. Assume that the

canonical map from HHX to ColimHHX?is an isomorphism. If each X?is resolvable, then X is

resolvable.

Theorem 1.2. Let X, >, and Z be connected simplicial sets of xnite p-type, and assume that Z is simply

connected. Let XPZ be a map of simplicial sets, and let >PZ be a Kan xbration. If X, >, and Z are

resolvable, then so is the xber product X??>.

These theorems allow us to argue inductively up towers of principal Kan "brations. The

following theorem proved in Section 6 provides a base case.

Theorem 1.3. K(Z/pZ,n) and K(Z??,n) are resolvable for n*1.

We conclude that every connected p-complete nilpotent simplicial set of "nite p-type is resolv-

able. The Main Theorem is now an elementary categorical consequence:

H?(X,>)?H?(X,UCH>)?h M E(CH>,CHX)

for X, > connected p-complete nilpotent simplicial sets of "nite p-type.

The proof of the Characterization Theorem is presented in Sections 7}10.

We mention here one more result in this paper. This result is needed in the proof of Theorem 1.3

but appears to be of independent interest. The work of May [22] provides the homology of

E?algebras in characteristic p with operations P? and ?P? (when p'2) for s3Z. It follows from

a check of the axioms and the identi"cation of ?P? as the Bockstein that when these operations are

applied to the F?-cochain complex of a simplicial set they perform the Steenrod operation of the

same names, where we understand P? to be the zero operation for s(0 and the identity for s"0.

The `algebra of all operationsa B therefore surjects onto the Steenrod algebra A with kernel

containing the two-sided ideal generated by 1!P?. The following theorem describes the precise

relationship between B and A.

Theorem 1.4. The left ideal of B generated by (1!P?) is a two-sided ideal whose quotient B/(1!P?)

is canonically isomorphic to A.

The analogue of the Main Theorem for "elds other than F M?is discussed in Appendix A. In

particular, we show that the analogue of the Main Theorem does not hold when F M?is replaced by

any "nite "eld.

A discussion of the composite UCH when the Main Theorem does not apply and a comparison

with p-pro-"nite completion is given in Appendix B (see also Remark 5.1).

M.A. Mandell / Topology 40 (2001) 43}94

47