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?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
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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
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M.A. Mandell / Topology 40 (2001) 43}94
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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$cients in F M?. We write CH(X;F?) and HH(X;F?) for the cochain complex and the
M.A. Mandell / Topology 40 (2001) 43}94
45
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cohomology of X with coe$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)
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M.A. Mandell / Topology 40 (2001) 43}94
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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
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2. The homotopy theory of E?algebras
In this section, wedevelop the homotopytheoretic resultswe need for the categoryof E-algebras.
In fact, the results of this section hold for the categories of algebras over the more general class of
operads described in Section 13. This class of operads includes all E?operads of di!erential graded
modules over a commutative ground ring. For convenience of notation for later reference, we state
everything in terms of the particular operad E of di!erential graded F M?-modules associated to the
given natural E?algebra structure on the cochain functor.
Although we do not prove that the category of E-algebras is a closed model category, the model
category framework provides a convenient language in which to present the results we need. We
assume familiarity with this language; we refer the unfamiliar reader to [9] for a good introduction
to model categories. In order to be able to use much of this language and in order to facilitate
constructions, we begin with the following well-known fact about categories of algebras over
operads of di!erential graded modules.
Proposition 2.1. The category of E-algebras is complete and cocomplete. Limits and xltered colimits
commute with the forgetful functor to diwerential graded modules.
The following de"nition speci"es the co"brations, "brations, and weak equivalences for our
model category results.
De5nition 2.2. We say that a map of E-algebras f:APB is a
(i) weak equivalence if it is a quasi-isomorphism.
(ii) xbration if it is a surjection.
(iii) coxbration if it has the left lifting property with respect to the acyclic "brations.
It is convenient to have a shorthand for indicating weak equivalences, "brations, and co"bra-
tions in diagrams. The following usage has become relatively standard.
Notation 2.3. The symbol `&a decorating an arrow indicates a map that is known to be or is
assumed to be a quasi-isomorphism. The arrow `?a indicates a map that is known to be or is
assumed to be a "bration. The arrow `Ma indicates a map that is known to be or is assumed to be
a co"bration.
We can identify the co"brations more intrinsically. In the following de"nition, for a di!eren-
tial graded module M, we denote by CM the cone on M; this is the di!erential graded module
whose underlying graded module is the sum of M and a copy of M shifted down, with di!erential
de"ned so that CM is contractible and the inclusion MPCM is a map of di!erential graded
modules.
De5nition 2.4. A map of E-algebras f:APB is relative cell inclusion if there exists a sequence of
E-algebra maps A"A?
??
P A?
??
P 2 such that
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M.A. Mandell / Topology 40 (2001) 43}94
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(i) B?Colimi?under A.
(ii) Each map i?is formed as a pushout of E-algebras
where ? denotes the free E-algebra functor, M???is a degreewise free di!erential graded module
with zero di!erential, CM???is the cone on M???, and the map M???PCM???is the canonical
inclusion.
We say that an E-algebra A is a cell E-algebra if the initial map F M?"E(0)PA is a relative cell
inclusion. A cell E-algebra A is xnite if each M?is "nitely generated and there is some N such that
M?"0 for n'N.
Clearly the relative cell inclusions are co"brations. The following proposition provides a near
converse.
Proposition 2.5. A map is a coxbration if and only if it is a retract of a relative cell inclusion.
The previous proposition is a formal consequence of a standard lift argument and the following
proposition that follows from an elementary application of the small objects argument.
Proposition 2.6. Any map of E-algebras f:APB can be factored functorially as f"p?i, where i is
a relative cell inclusion and p is an acyclic xbration.
We also mention the following lifting property. It follows by considering the left lifting property
for the relative cell inclusions ?F M?[n]P?CF M?[n], where F M?[n] denotes the degreewise free di!eren-
tial graded module with zero di!erential with one generator, in degree n.
Proposition 2.7. A map of E-algebras APB is an acyclic xbration if and only if it has the right lifting
property with respect to the coxbrations if and only if it has the right lifting property with respect to
coxbrations between cell E-algebras.
The previous two propositions give us one factorization property and one lifting property. We
cannot prove the other factorization and lifting properties in general. However, we can prove them
for co"brant E-algebras. We prove the following theorem in Section 13.
Theorem 2.8. Any map of E-algebras f:APB can be factored functorially as f"q?j, where j is
a relative cell inclusion that has the left lifting property with respect to the xbrations, and q is
a xbration. If A is coxbrant then j is in addition a quasi-isomorphism.
M.A. Mandell / Topology 40 (2001) 43}94
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Corollary2.9. Let A be coxbrant. Then a map of E-algebrasAPB is an acyclic coxbration if and only
if it has the left lifting property with respect to the xbrations.
Corollary 2.10. A map of E-algebras APB is a xbration if and only if it has the right lifting property
with respect to the acyclic coxbrations between cell E-algebras.
Using the fact that all E-algebras are "brant, the factorization and lifting properties above
provide su$cient tools to make the homotopy theory formalized by Quillen in [25] useful for
studying the category h M E, the localization of the category E obtained by formally inverting the
quasi-isomorphisms. Anticipating Theorem 2.13 below, we have already started referring to h M E as
the homotopy category of E-algebras; we now state the de"nition of homotopy.
De5nition 2.11. Let A be an E-algebra. A (Quillen) cylinder object for A is an E-algebra IA
equipped with maps ??,??:APIA and ?:IAPA such that ??#??:APAPIA is a co"bration,
? is a quasi-isomorphism,and the composite??(??#??) is the foldingmapAPAPA. We saythat
maps of E-algebras f?, f?:APB are (Quillen left) homotopic if there is a map f:IAPB such that
f?"f???and f?"f???; we call f a (Quillen left) homotopy from f?to f?. We denote by ?E(A,B) the
quotient of the mapping set E(A,B) by the equivalence relation generated by `homotopica.
In the case when A is a co"brant E-algebra, we can glue cylinder objects as in [25, Lemmas 1-3]
and see that `homotopica is already an equivalence relation on ?E(A,B).
Since our "brations are the surjections, the map ? is always an acyclic "bration, and so it is easy
to see that for arbitrary E-algebras A,B,C, composition in E induces an associative composition
?E(B,C)??E(A,B)P?E(A,C),
making ?E a category. The following proposition, the E-algebra analogue of the Whitehead
Theorem, is straightforward to deduce from the factorization and lifting properties above.
Proposition 2.12. Let A be a coxbrant E-algebra. A quasi-isomorphism of E-algebras BPC induces
a bijection ?(A,B)P?(A,C).
Since homotopic maps in E(A,B) represent the same map in h M E, the localization functor EPh M E
factors through the category ?E. Let ?E?denote the full subcategory of ?E consisting of the
co"brant E-algebras. We therefore obtain a functor ?E?Ph M E by restriction. The following
theorem, the analogue of Quillen [25, Theorems 1-1?], is now an immediate consequence of the
previous proposition.
Theorem 2.13. The functor ?E?Ph M E is an equivalence of categories. In particular h M E has small Hom
sets.
Another fundamental theorem that we can prove in this context is the analogue of [25, Theorem
4-3], needed for the construction of U in Section 4. The proof follows the standard one for model
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M.A. Mandell / Topology 40 (2001) 43}94
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simplicial set, a map XP> is a weak equivalence if and only if it induces a pro-isomorphism of
each homotopy pro-group ???X??P???>??.
The co"brations are the maps isomorphic to level maps that are level co"brations; in particular
all objectsare co"brant.It is shownthat the constantpro-simplicialseton a Kan simplicialset with
only "nitely many nontrivial homotopy groups is "brant in pro-S and a Kan "bration between
such simplicial sets is a "bration in pro-S. It follows that we can identify the functor
HH????X"HH(CH????X) as the set of maps from X to K(F M?,n) in the homotopy category of pro-S.
Thus, CH????converts co"brations to "brations and preserves weak equivalences. As an immediate
consequence of Theorems 2.14 and 2.15, we obtain the following proposition.
Proposition B.7. The (right) derived functor U?
h M E(A,CH????X)?pro-S(X,U?A).
of ;?
exists and gives an adjunction
The functor CH from the homotopy category to the homotopy category of E-algebras factors as
the composite of the constant functor and CH????, and so it follows that the functor U is the
composite of U?and the right-derived functor of Lim. The forgetful functor from Morel's model
category of pro-"nite simplicial sets to Isaksen's model category of pro simplicial sets is a right
adjoint that preserves "brations and acyclic "brations, and so the right-derived functor of Lim
from the homotopycategory of pro-"nite simplicial sets to the homotopycategory is the composite
of the right-derived functor of the forgetful functor and the right-derived functor of Lim from the
homotopy category of pro simplicial sets to the homotopy category. Since pro-"nite completion in
the sense of Sullivan is the composite of the completion functor from simplicial sets to pro-"nite
simplicial sets and the right-derived functor of Lim [24, Section 2.1], Theorem B.1 is an immediate
consequence of the following lemma.
Lemma B.8. Let X be a connected simplicial set. There is a xbrant pro-xnite simplicial set >, a weak
equivalence of pro-xnite simplicial sets X K P>, and a coxbrant approximation APCH????> such that
the map >P;?A is a weak equivalence of pro simplicial sets.
The remainder of the section is devoted to the proof of Lemma B.8. According to Morel [24,
Section 2.1], we can take >"?>?? to have the property that each >?is a connected `p-espace
xnisa, i.e. has "nitely many non-trivial homotopy groups, all of which are "nite p-groups. Choose
such a > and write I for the "ltering category opposite to the category that indexes >. It is not
hard to see that we can make an I diagram of co"brant E-algebras A?with a natural acyclic
"bration A?PCH>?and with the property that A"ColimA?is also co"brant. For example, it is
straightforward to check that ¸CH>?has this property where ¸ is the co"brant approximation
functor obtained by the small object argument in Proposition 2.6. Alternatively, after replacing
> with an isomorphic object if necessary, we can assume that I is a co"nite strongly directed
category, and then such a diagram A?is easily constructed by induction. Note that however the
A?are constructed, the induced map APCH????> is an acyclic "bration. We choose > and A in this
way in order to make the following observation.
Proposition B.9. For > and A as above, for each ?, the map from the constant pro simplicial set >?to
;?A?is a weak equivalence.
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Proof. According to Remark 7.4, since >?has only "nitely many nontrivial homotopy groups, all
of which are "nite p-groups, there is a "nite cell E-algebra B and a quasi-isomorphism BPCH>?.
By Proposition B.5, ;?B is isomorphic to the constant pro simplicial set on ;B, and so by the
Main Theorem, the map >?P;?B is a weak equivalence. But by the left lifting property, the map
BPCH>?can be factored through a quasi-isomorphism BPA?, and so the map >?P;?A?is also
a weak equivalence.
?
Let J be the category whose set of objects is the disjoint union of the sets of objects of the
R??where ? ranges over the objects of I. For a3R??, b3R??, we have a map aPb in J for each
map A?PA?in I that maps the pair of di!erential graded submodules (M,N) corresponding to
a into the pair of di!erential graded submodules corresponding to b. Clearly J is a "ltered
category. The functors D??:R??PE assemble to a functor D:JPE, which we regard as an
element of ind-E. We have a canonical map DP?A?? covering the forgetful functor JPI and
inducing an isomorphism ColimDPColimA?"A. Since D is a diagram of compact E-algebras,
the map DP?A?? factors through an isomorphism DPcA by Proposition B.6.
Proof of Lemma B.8. If we choose a basepoint for X, we obtain compatible basepoints for the
>?so that > is a system of based connected simplicial sets. Then it su$ces to show that the map
>P;?A inducesa pro-isomorphismof eachhomotopy pro-group??>P??;?A. By construction,
the map >P;?A factors through the map >P?;A??; we base ;A?and the simplicial sets in
;?A?at the image of the basepointof >?.Looking at D, wecan identify??;?A as the limit (over ? in
I) in pro-groups of the pro-groups ???;?A??. Since ??> is the limit (over ? in I) in pro-groups of
the constant pro-groups ??>?, the lemma now follows from Proposition B.9.
?
Appendix C. E?ring spectra under HF M?
We sketch how the arguments in this paper can be modi"ed to prove the following unpublished
theorem of Dwyer and Hopkins [8] comparing the p-adic homotopy category with the homotopy
category of E?HF M?ring spectra.
Theorem C.1 (Dwyer}Hopkins). The free mapping spectrum functor F((!)?,F M?) induces an equiva-
lence between the homotopy category of connected p-complete nilpotent spaces of xnite p-type and
a full subcategory of the homotopy category of E?HF M?ring spectra.
By the homotopy category of E?HF M?ring spectra, we mean the category obtained from the
category of E?ring spectra under the (co"brant) E?ring spectrum HF M?by formally inverting the
weak equivalences. The free mapping spectrum F(X?,HF M?) is naturally an E?ring spectrum with
an E?ring map
HF M?"F(*?,HF M?)PF(X?,HF M?)
induced by the collapse map XP*. The functor F((!)?,F M?) therefore takes values in the category
of E?HF M?ring spectra. This functor is the spectrum analogue of the singular chain complex. Its
right derived functor represents unreduced ordinary cohomology with coe$cients in F M?in the
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sense that there is a canonical map ??H(F(X?,F M?))PHH(X;F M?) that is an isomorphism if X is
a CW complex.
It is convenient for us to use a modern variant of the category of E?HF M?ring spectra, the
category of commutative HF M?-algebras, a certain subcategory introduced in [10]. The forgetful
functor from commutative HF M?-algebras to E?HF M?ring spectra induces an equivalence of
homotopycategories.We have a commutativeHF M?-algebravariationof the free mappingspectrum
functor, given by
FX"S?LF(X?,HF M?).
There is a natural map FXPF(X?,HF M?) that is always a weak equivalence, and so it su$ces to
prove that the functor F induces an equivalence between the homotopy category of connected
p-complete nilpotent spaces of "nite p-type and a full subcategory of the homotopy category of
commutative HF M?-algebras. We denote the category of commutative HF M?-algebras as C. By [10,
VII.4.10], C is a closed model category with weak equivalences the weak equivalences of the
underlying spectra; we denote its homotopy as h M C.
The commutative HF M?-algebra FX is the `cotensora of HF M?with X [10, VII.2.9]. In general, the
cotensor A? of a commutative HF M?-algebra A with the space X is the commutative HF M?-algebra
that solves the universal mapping problem C(!,A?)?U(X,C(!,A)), where U denotes the
category of (compactly generated and weakly Hausdor!) spaces. Similarly, the tensor A?X of
A with the space X is the commutative HF M?-algebra that solves the universal mapping problem
C(A?X,!)?U(X,C(A,!)). Clearly, when they exist, A? and A?X are unique up to canonical
isomorphism, and [10, VII.2.9] guarantees that they exist for any A and any X. The signi"cance of
the identi"cation of FX as the cotensor is in the following proposition.
Proposition C.2. The functor ¹:CPU dexned by ¹A"C(A,HF M?) is a continuous contravariant
rightadjoint to F. In other words,there is a homeomorphismU(X,¹A)?C(A,FX),naturalin the space
X and the commutative HF M?-algebra A.
We have introduced the notion of tensor here to take advantage of Elmendorf et al. [10,
VII.4.16] that identi"es the tensor A?I as a Quillen cylinder object when A is co"brant. This
allows us to relate the homotopies in the sense of Quillen with topological homotopies de"ned in
terms of (!)?I or in terms of paths in mapping spaces. In particular, since all objects in C are
"brant, it follows that the natural transformation ??(C(A,!))Ph M C(A,!) is an isomorphism
when A is co"brant. Since the adjunctionisomorphism U(X,¹A)?C(A,FX) is a homeomorphism,
letting X vary over the spheres, we obtain the following proposition.
Proposition C.3. The functor T preserves weak equivalences between coxbrant objects.
As a slight generalization of the proof of Elmendorf et al. [10, VII.4.16], it is elementary to check
that when A is a co"brant objectof C and APB is a co"bration,the map (A?I)P?BPB?I is an
acyclic co"bration and therefore (since every object is "brant) the inclusion of a retract. Since
¹ also converts pushouts to pullbacks, applying ¹ and using the tensor adjunction, we obtain the
following proposition.
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Proposition C.4. The functor T converts coxbrations to xbrations.
The functors F and ¹ are therefore a model category adjunction. In particular, we obtain the
following proposition.
Proposition C.5. The (right) derived functors F and T of F and T exist and give a contravariant right
adjunction H?(X,TA)?h M C(A,FX).
For the purposes of this section, let us say that a space X is HF M?-resolvable if the unit of the
derived adjunction XPTFX is a weak equivalence.Thus, we need to show that if X is a connected
p-complete nilpotent space of "nite p-type, then X is HF M?-resolvable. Again, we work by induction
up principally re"ned Postnikov towers. The following analogue of Theorem 1.1 can be proved
from Proposition C.4 by essentially the same argument used to prove Theorem 1.1 from Proposi-
tion 4.4.
Proposition C.6. Let X"LimX?be the limit of a tower of Serre xbrations. Assume that the canonical
map from HHX to ColimHHX?is an isomorphism. If each X?is HF M?-resolvable, then X is resolvable.
We have in addition the following analogue of Theorem 1.2.
Theorem C.7. Let X, >, and Z be connected spaces of xnite p-type, and assume that Z is simply
connected. Let XPZ be a map, and let >PZ be a Serre xbration. If X, >, and Z are HF M?-resolvable,
then so is the xber product X??>.
The proof of this theorem is essentially the same in outline as the proof of Theorem 1.2. The
analogue of Lemma 5.2 can be proved by observing that the bar construction of the co"brant
approximations in C is equivalent to the (thickened) realization of F applied to the cobar
construction of the singular simplicial sets on the spaces X?, >?, and Z?. Some "ddling with the
"ltration induced by the cosimplicial direction of the cobar construction and the "ltration induced
by the skeletal "ltration of the singular simplicial sets allows the identi"cation of
Tor?????
?H(F?X??,F?>??) as TorH?H?(CHX,CH>) and the composite map
TorH?H?(CHX,CH>)?Tor?????
as the Eilenberg}Moore map.
To complete the proof of Theorem C.1, we need to see that K(Z/pZ,n) is HF M?-resolvable. It then
follows as in Section 1.3 that K(Z??,n) is HF M?-resolvable and by induction up principal Postnikov
towers that every connected p-complete nilpotent space of "nite p-type is HF M?-resolvable. The
remainder of the appendix is devoted to sketching a proof of the following theorem.
?H(F?X??,F?>??)P??HF(?X???
?>??)?HH(X??>)
Theorem C.8. For n*1, K(Z/pZ,n) is HF M?-resolvable.
The homotopy groups of a commutative HF M?-algebra have an action by the algebra B, and it is
elementary to show that for the `freea commutative HF M?-algebra on the spectrum S??, denoted
?S??
?F M?in [10], ??H?S??
?F M?is ? M B??
?, the extended F M?-algebra on the enveloping algebra of the free
M.A. Mandell / Topology 40 (2001) 43}94
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unstable B-module on one generator in degree n. We construct a commutative HF M?-algebra B?as
the commutative HF M?-algebra that makes the following diagram a pushout in C:
Here p?is any map in the unique homotopy class that on homotopy groups sends the fundamental
class of
???S??
?F M?
a:?S??
homotopy ?CS??
map B?PFK(Z/pZ,n).
to1!P?
appliedto the fundamentalclass.Choosingamap
F M?PFK(Z/pZ,n) that represents the fundamental class of H?(K(Z/pZ,n)), and a null
F M?PFK(Z/pZ,n) for the map p??a:?S??
F M?PFK(Z/pZ,n), we obtain an induced
Proposition C.9. For n*1, the map B?PFK(Z/pZ,n) is a weak equivalence.
The proof uses the Eilenberg}Moorespectral sequenceof Elmendorfet al. [10, IV.4.1] in place of
the Eilenberg}Moore spectral sequence of Section 3, but otherwise is the same as the proof of
Theorem 6.2.
Since B?is a co"brant commutative HF M?-algebra, the unit of the derived adjunction is represent-
ed by the map K(Z/pZ,n)P¹B?adjoint to the map constructed above. Since B?is de"ned as
a pushout of a co"bration, Proposition C.4 allows us to identify ¹B?as the pullback of "bration.
Looking at the mapping spaces and using the freeness adjunction, we see that ¹B?is the homotopy
"ber of an endomorphism on K(F M?,n). Write ??for the induced endomorphism on F M?. To see that
¹B?is a K(Z/pZ,n), it su$ces to show that ??is 1!?. Once we know that ¹B?is a K(Z/pZ,n), the
argumentof Corollary6.3 shows that the map K(Z/pZ,n)P¹B?is a weak equivalence,completing
the proof of Theorem C.8.
Unfortunately, the simple argument given in Proposition 6.5 to identify ??as 1!? in the
algebraic case does not have a topological analogue. Here we must use homotopy theory to make
this identi"cation. The key observation is that the commutative HF M?-algebras B?are related by
`suspensiona.We make this precisein the followingproposition.For this proposition,note that the
de"nition of B?makes sense for n"0, although the map B?PFK(Z/pZ,0) may not be a weak
equivalence.
Proposition C.10. For n'0, B???is homotopy equivalent as a commutative HF M?-algebra to the
pushout of the following diagram:
where the map B?PHF M?is the augmentation B?PFK(Z/pZ,n)PF*"HF M?induced by the inclu-
sion of the basepoint of K(Z/pZ,n) and the map B?PB?S? is induced by the inclusion*PS?.
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For an augmented commutative HF M?-algebra A, denote the analogous pushout for A as ?CA. If
we give ?S??
?F M?the augmentation induced by applying ? to the map S??F M?P*, then ?C?S??
canonically isomorphic to ?S????
?F M?
. This gives us a canonical suspension homomorphism
?:? ???AP? ??????CA, where ? ?His the kernel of the augmentation map ?HAP?HHF M?. The
following proposition is closely related to and can be deduced from [22, 3.3].
?F M?is
Proposition C.11. The suspension homomorphism ? commutes with the operation P? for all s.
We can choose the map p?in the construction of B?to be augmented for the augmentation
described on ?S??F M?above. Then it follows from the previous proposition that ?Cp?is homotopic to
p???. This observation can be used to prove Proposition C.10.
It follows from Proposition C.10 that ¹B???is the loop space of ¹B?. In fact, we see from the
discussion above that the "ber sequence for ¹B???
¹B???PK(F M?,n!1)PK(F M?,n!1)
is the loop of the corresponding "ber sequence for ¹B?. In particular, ??and ????are the same
endomorphisms of F M?. Since P? performs the pth power map on classes in degree zero, ??is 1!?.
We conclude that ??is 1!?.
15. For further reading
The following references are also of interest to the reader: [2}6, 14, 28, 29].
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