E1 ALGEBRAS AND pADIC HOMOTOPY THEORY
MICHAEL A. MANDELL
Abstract.Let Fpdenote the field with p elements and Fpits algebraic clo
sure. We show that the singular cochain functor with coefficients in Fpi*
*nduces
a contravariant equivalence between the homotopy category of connected p
complete nilpotent spaces of finite ptype and a full subcategory of the*
* homo
topy category of E1 Fpalgebras. Draft: October 30, 1998, 20:22
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, Sulliv*
*an,
and others showed that, viewed rationally, homotopy theory becomes completely
algebraic. In particular, Sullivan showed that an important subcategory of the
homotopy category of rational spaces is contravariantly equivalent to a subcate*
*gory
of the homotopy category of commutative differential graded Qalgebras, and that
the functor underlying this equivalence is closely related to the singular coch*
*ain
functor. In this paper, we offer a similar theorem for padic homotopy theory.
Since the noncommutativity of the multiplication of the Fp singular cochains*
* is
visible already on the homology level in the Steenrod operations, one would not*
* ex
pect that any useful subcategory of the padic homotopy category to be equivale*
*nt
to a category of commutative differential graded algebras. We must instead look*
* to
a more sophisticated class of algebras, the E1 algebras [18]. E1 algebras, rou*
*ghly,
are differential graded modules with an infinitely coherent homotopy associative
and commutative multiplication. They provide a generalization of commutative
differential graded algebras that admits homology operations as commutativity o*
*b
structions generalizing the Steenrod operations. To capture padic homotopy the
ory, even the category of E1 Fpalgebras is not quite sufficient; rather we co*
*nsider
E1 algebras over the algebraic closure Fp of Fp. We prove the following theore*
*m.
Main Theorem. The singular cochain functor with coefficients in Fp induces
a contravariant equivalence from the homotopy category of connected pcomplete
nilpotent spaces of finite ptype to a full subcategory of the homotopy categor*
*y of
E1 Fpalgebras.
The homotopy category of connected pcomplete nilpotent spaces of finite pty*
*pe
is a full subcategory of the padic homotopy category, the category obtained fr*
*om
the category of spaces by formally inverting those maps that induce isomorphisms
on singular homology with coefficients in Fp. The padic homotopy category it
self can be regarded as a full subcategory of the homotopy category, the catego*
*ry
____________
Date: October 30, 1998, 20:22.
1991 Mathematics Subject Classification. Primary 55P15; Secondary 55P60.
This research was supported in part by an Alfred P. Sloan Dissertation Fello*
*wship.
1
2 MICHAEL A. MANDELL
obtained from the category of spaces by formally inverting the weak equivalence*
*s.
We remind the reader that a connected space is pcomplete, nilpotent, and of fi*
*nite
ptype if and only if its Postnikov tower has a principal refinement in which e*
*ach
fiber is of type K(Z=pZ ; n) or K(Z^p; n), where Z^pdenotes the padic integers.
By the homotopy category of E1 Fpalgebras, we mean the category obtained
from the category of algebras over a particular but unspecified E1 Fp operad by
formally inverting the maps in that category that are quasiisomorphisms of the
underlying differential graded Fpmodules, the maps that induce an isomorphism
of homology groups. It is wellknown that up to equivalence, this category does*
* not
depend on the operad chosen. We refer the reader to [18, I] for a good introduc*
*tion
to operads, E1 operads, and E1 algebras.
To complete the picture, we need to identify intrinsically the subcategory of*
* the
homotopy category of E1 Fpalgebras that the Main Theorem asserts an equiva
lence with. Although we can write a necessary and sufficient condition for an E1
Fpalgebra to be quasiisomorphic to the singular cochain complex of a connecte*
*d p
complete nilpotent space of finite ptype, it is relatively unenlightening and *
*difficult
to verify in practice. This condition is stated precisely in Section 7 and is e*
*ssentially
the E1 Fpalgebra analogue of the existence of a finite ptype principal Postn*
*ikov
tower. Unsurprisingly, restricting consideration to simply connected spaces mak*
*es
the identification significantly easier. In fact, we can write necessary and su*
*fficient
conditions for an E1 Fpalgebra to be quasiisomorphic to the singular cochain
complex of a 1connected space of finite ptype in terms of its homology and the
generalized Steenrod operation P 0.
Characterization Theorem. An E1 differential graded Fpalgebra A is quasi
isomorphic in the category of E1 Fpalgebras to the singular cochain complex o*
*f a
1connected (pcomplete) space of finite ptype if and only if HiA is zero for *
*i < 0,
H0A = Fp , H1A = 0, and for i > 1, HiA is finite dimensional over Fp and
generated as an Fpmodule by the fixed points of the operation P 0.
Succinctly, the Characterization Theorem states that an E1 Fpalgebra A is
quasiisomorphic to the singular cochain complex of a 1connected space of fini*
*te
ptype if and only if the homology of A looks like the cohomology of such a spa*
*ce
as a module over the generalized Steenrod algebra.
Comparison with Other Approaches. The papers [13, 17, 26] and the un
published ideas of [8] all compare padic homotopy theory to various homotopy
categories of algebras (or coalgebras). We give a short comparison of these res*
*ults
to the results proved here.
The first announced results along the lines of our Main Theorem appeared in
[26]. The arguments there are not well justified, however, and some of the resu*
*lts
appear to be wrong.
More recently, [13, 17] have compared the padic homotopy category with the
homotopy categories of simplicial cocommutative coalgebras and cosimplicial com
mutative algebras. In particular, [13] proves that the padic homotopy category
embeds as a full subcategory of the homotopy category of cocommutative simpli
cial Fpcoalgebras. The analogue of the Characterization Theorem is not known in
this context. It is straightforward to describe the relationship between the re*
*sults
of [13] and our Main Theorem. There is a functor from the homotopy category
of simplicial cocommutative coalgebras to the homotopy category of E1 algebras
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 3
given by normalization of the dual cosimplicial commutative algebra [15] (see a*
*lso x1
below). Applied to the singular simplicial chains of a space, we obtain the sin*
*gular
cochain complex of that space. Our Main Theorem implies that on the subcategory
of nilpotent spaces of finite ptype, this refined functor remains a full embed*
*ding.
This gives an affirmative answer to the question asked in [17, 6.3].
The unpublished ideas of [8] for comparing the padic homotopy category to the
homotopy category of E1 ring spectra under the EilenbergMac Lane spectrum
HFp , would give a "brave new algebra" version of our Main Theorem. A proof of
such a comparison can be given along similar lines to the proof of our Main The*
*o
rem. We sketch the argument in Appendix C. The analogue of the Characterization
Theorem in this context was not considered in [8], but can be proved by essenti*
*ally
the same arguments as the proof of our Characterization Theorem. A direct com
parison between our approach and this approach to padic homotopy theory would
require a comparison of the homotopy category of E1 HFp ring spectra and the
category of E1 Fpalgebras, and also an identification of the composite functor
from spaces to E1 differential graded Fpalgebras as the singular cochain func*
*tor.
We will provide this comparison and this identification in [19] and [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 differential graded modules "cohomologically" with the
differential raising degrees. This makes the cochain complexes concentrated in
nonnegative degrees, but forces E1 operads to be concentrated in nonpositive
degrees. Along with this convention, we write the homology of a differential gr*
*aded
module M as H*M. We work almost exclusively with ground ring Fp; throughout
this paper, C*X and H*X always denote the cochain complex and the cohomology
of X taken with coefficients in Fp. We write C*(X; Fp) and H*(X; Fp) for the
cochain complex and the cohomology of X with coefficients in Fp or C*(X; k) and
H*(X; k) for these with coefficients in a commutative ring k.
The first prerequisite to the Main Theorem is recognizing that the singular
cochain functor can be regarded as a functor into the category of Ealgebras for
some E1 Fpoperad E. In fact, for the purpose of this paper, the exact constru*
*c
tion of this structure does not matter so long as the (normalized) cochain comp*
*lex
of a simplicial set is naturally an Ealgebra. 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 alge*
*bra",
an algebra over an acyclic operad Z, the "EilenbergZilber" operad. The operad
Z is not an E1 operad however since it is not free and since it is nonzero in
both positive and negative degrees. To fix this, let Z be the "(co)connective *
*cover"
of Z: Z (n) is the differential graded Fpmodule that is equal to Z(n) in degre*
*es
less than zero, equal to the kernel of the differential in degree zero, and zer*
*o in
positive degrees. The operadic multiplication of Z lifts to Z, making it an acy*
*clic
operad. Tensoring Z with an E1 operad C gives an E1 operad E and a map of
operads E ! Z. The cochain complex of a simplicial set then obtains the natural
structure of an algebra over the E1 operad E.
We write E for the category of Ealgebras. Since we are assuming that the fun*
*c
tor C* from spaces to Ealgebras factors through the category of simplicial set*
*s, we
4 MICHAEL A. MANDELL
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 ho
motopy category; this category is equivalent to the category of Kan complexes a*
*nd
homotopy classes of maps and to the category of CW spaces and homotopy classes
of maps. Since the cochain functor converts Fphomology isomorphisms and in
particular weak equivalences of simplicial sets to quasiisomorphisms of Ealge*
*bras,
the (total) derived functor exists as a contravariant functor from the homotopy
category to the homotopy category of Ealgebras. We prove the Main Theorem
by constructing a right adjoint U from the homotopy category of Ealgebras 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 UC*, we need
some tools to help us understand the homotopy category of Ealgebras. The tools
we need are precisely those provided by Quillen's theory of closed model catego*
*ries
[25] (see also [9]). Unfortunately, we have not been able to verify that the ca*
*tegory
of Ealgebras is a model category. Nevertheless, the category of Ealgebras 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 The
orem require understanding of the derived coproduct and the homotopy pushout
of Ealgebras. We summarize the results we need in Section 3; the proofs of the*
*se
results are in Section 14.
We construct in Section 4 a contravariant functor U from the category of E
algebras to the category of simplicial sets that is the right adjoint to C*. O*
*ur
model theoretic results allow us to show that the right derived functor of U ex
ists and is right adjoint to the derived functor of C*; this derived functor is*
* our
functor U mentioned above. Precisely, U is a contravariant functor from the ho
motopy category of Ealgebras to the homotopy category, and we have a canonical
isomorphism
H o(X; UA) ~=hE (A; C*X)
for a simplicial set X and an Ealgebra A. Here and elsewhere H o denotes the
homotopy category and hE denotes the homotopy category of Ealgebras.
We write uX for the "unit" of the derived adjunction X ! UC*X. For the pur
poses of this paper, we say that a simplicial set X is resolvable by E1 Fpalg*
*ebras
or just resolvable if the map uX 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 fibrations . .!.Xn ! . .X.0.
Assume that the canonical map from H*X to ColimH*Xn is an isomorphism. If
each Xn is resolvable, then X is resolvable.
Theorem 1.2. Let X, Y , and Z be connected simplicial sets of finite ptype, a*
*nd
assume that Z is simply connected. Let X ! Z be a map of simplicial sets, and
let Y ! Z be a Kan fibration. If X, Y , and Z are resolvable, then so is the fi*
*ber
product X xZ Y .
These theorems allow us to argue inductively up towers of principal Kan fibra
tions. The following theorem proved in Section 6 provides a base case.
Theorem 1.3. K(Z=pZ ; n) and K(Z^p; n) are resolvable for n 1.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 5
We conclude that every connected pcomplete nilpotent simplicial set of finite
ptype is resolvable. The Main Theorem is now an elementary categorical conse
quence:
H o(X; Y ) ~=H o(X; UC*Y ) ~=hE (C*Y; C*X)
for X, Y connected pcomplete nilpotent simplicial sets of finite ptype.
The proof of the Characterization Theorem is presented in Sections 710.
We mention here one more result in this paper. This result is needed in the p*
*roof
of Theorem 1.3 but appears to be of independent interest. The work of [22] prov*
*ides
the homology of E1 algebras in characteristic p with operations P sand fiP s(wh*
*en
p > 2) for s 2 Z. It follows from a check of the axioms and the identification *
*of fiP 0
as the Bockstein that when these operations are applied to the Fpcochain compl*
*ex
of a simplicial set they perform the Steenrod operation of the same names, where
we understand P sto be the zero operation for s < 0 and the identity for s = 0.*
* The
"algebra of all operations" B therefore surjects onto the Steenrod algebra A wi*
*th
kernel containing the twosided ideal generated by 1  P 0. The following theor*
*em
describes the precise relationship between B and A.
Theorem 1.4. The left ideal of B generated by (1P 0) is a twosided ideal who*
*se
quotient B=(1  P 0) is canonically isomorphic to A.
The analogue of the Main Theorem for fields other than Fp is discussed in
Appendix A. In particular, we show that the analogue of the Main Theorem does
not hold when Fp is replaced by any finite field.
A discussion of the composite UC* when the Main Theorem does not apply
and a comparison with pprofinite completion is given in Appendix B (see also
Remark 5.1).
2.The Homotopy Theory of E1 Algebras
In this section we develop the homotopy theoretic results we need for the cat*
*egory
of Ealgebras. 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 o*
*perads
includes all E1 operads of differential graded modules over a commutative grou*
*nd
ring. For convenience of notation for later reference, we state everything in t*
*erms
of the particular operad E of differential graded Fpmodules associated to the *
*given
natural E1 algebra structure on the cochain functor.
Although we do not prove that the category of Ealgebras 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 re*
*fer
the unfamiliar reader to [9] for a good introduction to model categories. In or*
*der
to be able to use much of this language and in order to facilitate construction*
*s, we
begin with the following wellknown fact about categories of algebras over oper*
*ads
of differential graded modules.
Proposition 2.1.The category of Ealgebras is complete and cocomplete. Lim
its and filtered colimits commute with the forgetful functor to differential gr*
*aded
modules.
The following definition specifies the cofibrations, fibrations, and weak equ*
*iva
lences for our model category results.
Definition 2.2.We say that a map of Ealgebras f :A ! B is a:
6 MICHAEL A. MANDELL
(i)weak equivalence if it is a quasiisomorphism.
(ii)fibration if it is a surjection.
(iii)cofibration if it has the left lifting property with respect to the acycl*
*ic fibra
tions.
It is convenient to have a shorthand for indicating weak equivalences, fibrat*
*ions,
and cofibrations in diagrams. The following usage has become relatively standar*
*d.
Notation 2.3. The symbol "~" decorating an arrow indicates a map that is known
to be or is assumed to be a quasiisomorphism. The arrow "i" indicates a map
that is known to be or is assumed to be a fibration. The arrow "ae" indicates a
map that is known to be or is assumed to be a cofibration.
We can identify the cofibrations more intrinsically. In the following definit*
*ion,
for a differential graded module M, we denote by CM the cone on M; this is the
differential graded module whose underlying graded module is the sum of M and
a copy of M shifted down, with differential defined so that CM is contractible *
*and
the inclusion M ! CM is a map of differential graded modules.
Definition 2.4.A map of Ealgebras f :A ! B is relative cell inclusion if there
exists a sequence of Ealgebra maps A = A0 i0!A1 i1!. .s.uch that
(i)B ~=Colimin under A.
(ii)Each map in is formed as a pushout of Ealgebras
EMn _____//ECMn+1
 
 
fflffl fflffl
An ___in__//_An+1
where E denotes the free Ealgebra functor, Mn+1 is a degreewise free diff*
*er
ential graded module with zero differential, CMn+1 is the cone on Mn+1, and
the map Mn+1 ! CMn+1 is the canonical inclusion.
We say that an Ealgebra A is a cell Ealgebra if the initial map Fp = E(0) ! A*
* is
a relative cell inclusion. A cell Ealgebra A is finite if each Mn is finitely *
*generated
and there is some N such that Mn = 0 for n > N.
Clearly the relative cell inclusions are cofibrations. The following proposi*
*tion
provides a near converse.
Proposition 2.5.A map is a cofibration if and only if it is a the 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 Ealgebras f :A ! B can be factored functorially
as f = p O i, where i is a relative cell inclusion and p is an acyclic fibratio*
*n.
We also mention the following lifting property. It follows by considering the
left lifting property for the relative cell inclusions EFp [n] ! ECFp [n], wher*
*e Fp[n]
denotes the degreewise free differential graded module with zero differential w*
*ith
one generator, in degree n.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 7
Proposition 2.7.A map of Ealgebras A ! B is an acyclic fibration if and only
if it has the right lifting property with respect to the cofibrations if and on*
*ly if it has
the right lifting property with respect to cofibrations between cell Ealgebras.
The previous two propositions give us one factorization property and one lift*
*ing
property. We cannot prove the other factorization and lifting properties in gen*
*eral.
However, we can prove them for cofibrant Ealgebras. We prove the following
theorem in Section 13.
Theorem 2.8. Any map of Ealgebras f :A ! B can be factored functorially as
f = q O j, where j is a relative cell inclusion that has the left lifting prope*
*rty with
respect to the fibrations, and q is a fibration. If A is cofibrant then j is in*
* addition
a quasiisomorphism.
Corollary 2.9.Let A be cofibrant. Then a map of Ealgebras A ! B is an acyclic
cofibration if and only if it has the left lifting property with respect to the*
* fibrations.
Corollary 2.10.A map of Ealgebras A ! B is a fibration if and only if it has t*
*he
right lifting property with respect to the acyclic cofibrations between cell E*
*algebras.
Using the fact that all Ealgebras are fibrant, the factorization and lifting*
* prop
erties above provide sufficient tools to make the homotopy theory formalized by
Quillen in [25] useful for studying the category hE , the localization of the c*
*ate
gory E obtained by formally inverting the quasiisomorphisms. Anticipating The
orem 2.13 below, we have already started refering to hE as the homotopy category
of Ealgebras; we now state the definition of homotopy.
Definition 2.11.Let A be an Ealgebra. A (Quillen) cylinder object for A is
an Ealgebra IA equipped with maps @0; @1: A ! IA and oe :IA ! A such that
@0+ @1: A q A ! IA is a cofibration, oe is a quasiisomorphism, and the composi*
*te
oe O (@0 + @1) is the folding map A q A ! A. We say that maps of Ealgebras
f0; f1: A ! B are (Quillen left) homotopic if there is a map f :IA ! B such that
f0 = f O @0 and f1 = f O @1; we call f a (Quillen left) homotopy from f0 to f1.*
* We
denote by ssE (A; B) the quotient of the mapping set E (A; B) by the equivalence
relation generated by "homotopic".
In the case when A is a cofibrant Ealgebra, we can glue cylinder objects as *
*in
[25, Lemma 13] and see that "homotopic" is already an equivalence relation on
ssE (A; B).
Since our fibrations are the surjections, the map oe is always an acyclic fib*
*ration,
and so it is easy to see that for arbitrary Ealgebras A; B; C, composition in E
induces an associative composition
ssE (B; C) x ssE (A; B) ! ssE (A; C);
making ssE a category. The following proposition, the Ealgebra analogue of the
Whitehead Theorem, is straightforward to deduce from the factorization and lift*
*ing
properties above.
Proposition 2.12.Let A be a cofibrant Ealgebra. A quasiisomorphism of E
algebras B ! C induces a bijection ss(A; B) ! ss(A; C).
Since homotopic maps in E (A; B) represent the same map in hE , the localizat*
*ion
functor E ! hE factors through the category ssE . Let ssEc denote the full subc*
*at
egory of ssE consisting of the cofibrant Ealgebras. We therefore obtain a func*
*tor
8 MICHAEL A. MANDELL
ssEc ! hE by restriction. The following theorem, the analogue of [25, Theorem
11'], is now an immediate consequence of the previous proposition.
Theorem 2.13. The functor ssEc ! hE is an equivalence of categories. In partic
ular hE has small Hom sets.
Another fundamental theorem that we can prove in this context is the analogue*
* of
[25, Theorem 43], needed for the construction of U in Section 4. The proof fol*
*lows
the standard one for model categories with only minor modifications and is left*
* to
the reader. Although we apply it to a contravariant functor in the construction*
* of
U, we write it here in the familiar covariant form because this gives a much cl*
*earer
and unambiguous statement.
Theorem 2.14. Let L: E ! M and R: M ! E be left and right adjoints between
the category of Ealgebras E and a closed model category M .
(i)If L preserves cofibrations between cofibrant objects and R preserves fibr*
*ations,
then the left derived functor of L and the right derived functor of R exis*
*t and
are adjoint. Moreover, L converts quasiisomorphisms between cofibrant E
algebras to weak equivalences, and the restriction of the left derived fun*
*ctor
of L to the cofibrant Ealgebras is naturally isomorphic to the derived fu*
*nctor
of the restriction of L.
(ii)Suppose that (i) holds and in addition for any cofibrant Ealgebra A and a*
*ny
fibrant object Y in M , a map A ! RY is a quasiisomorphism if and only if
the adjoint LA ! Y is a weak equivalence. Then the left derived functor of
L and the right derived functor of R are inverse equivalences.
The lifting properties provide the following useful alternative hypotheses.
Theorem 2.15. The hypothesis of 2.14.(i) is equivalent to each of the followin*
*g.
(i)L preserves cofibrations between cofibrant objects and acyclic cofibration*
*s be
tween cofibrant objects.
(ii)R preserves fibrations and acyclic fibrations.
We note here for future reference that the analogues of the previous two theo*
*rems
also hold when E or M (or both) is replaced by an undercategory A=E for a
cofibrant Ealgebra A.
3.The E1 Torsion Product
We use the results on the homotopy theory of Ealgebras of the last section to
study coproducts and homotopy pushouts of Ealgebras in this section. We show
that the homology of these is closely related to the differential torsion produ*
*ct. We
state the results in this section for the category of algebras over the operad *
*E of
differential graded Fpmodules, but except as noted they apply more generally to
the category of algebras over any E1 operad of differential graded modules ove*
*r a
commutative ring.
Definition 3.1.Let A ! B and A ! C be maps of Ealgebras. We define the
E1 torsion product of B and C under A by
E1 Tor*A(B; C) = H*(B0qA0C0)
where A0 is a cofibrant approximation of A and B0 and C0 are cofibrant approx
imations of B and C in the category of Ealgebras under A0, i.e. A0 is cofibra*
*nt
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 9
and we have the following commutative diagrams with arrows quasiisomorphisms,
fibrations, and cofibrations as indicated.
B0oo___oA0o//__//C0
  
~  ~  ~
fflfflfflfflfflfflfflfflfflfflfflffl
B oo____A _____//_C
We omit A from the notation, writing E1 Tor*(B; C), when A is the initial objec*
*t.
The standard lift and homotopy arguments combined with the following theorem
imply that the E1 torsion product is welldefined and that it only depends on *
*the
diagram B A ! C in the homotopy category of diagrams of this form.
Theorem 3.2. Given the following diagram of Ealgebras, with the vertical maps
quasiisomorphisms and the righthand horizontal maps cofibrations
B0oo___ A0//___//C0
~  ~  ~
fflffl fflfflfflffl
B oo____A //___//C;
if A, A0, B, and B0 are cofibrant, then the induced map of pushouts
B0qA0C0! B qA C
is a quasiisomorphism.
Proof.Factor the map A0 ! B0 as a cofibration A0 ! B00followed by a quasi
isomorphism B00! B0; it suffices to show that the induced maps B00qA0 C0 !
B0qA0 C0 and B00qA0 C0 ! B qA C are both quasiisomorphisms. As noted at
the close of the last section, Theorems 2.14 and 2.15 hold with E and M replaced
with the undercategories X=E and Y=E when X and Y are cofibrant Ealgebras.
Applying this for the adjoint pair of functors induced by a map X ! Y and apply*
*ing
the argument for K. Brown's lemma [9, 9.9], it follows that for cofibrant Ealg*
*ebras,
the pushout of a weak equivalence along a cofibration is a weak equivalence. The
theorem then follows by noting that the map B00qA0C0! B0qA0C0is the pushout
of a weak equivalence along a cofibration and the map B00qA0C0 ! B qA C can *
* __
be factored as a sequence of pushouts of weak equivalences along cofibrations. *
* __
Corollary 3.3.Let A, B be cofibrant Ealgebras, A ! B a map of Ealgebras and
A ! C a cofibration of Ealgebras. Then the canonical map E1 Tor*A(B; C) !
H*(B qA C) is an isomorphism.
The next theorem compares the E1 torsion product to the ordinary differential
torsion product over the ground ring. Since our differential graded modules are
integer graded, we should say a few words about what we mean by the differential
torsion product. For differential graded modules M; N, let Tor*(M; N) be the
homology of the left derived functor of the bifunctor () () on M; N. This
derived functor is proved to exist for example in [18, xIII.4], and it coincide*
*s with
the left derived functor obtained by fixing one of the variables M or N. We hav*
*e a
canonical map Tor*(M; N) ! H*M H*N, which is an isomorphism in the case
of main interest since Fp is a field.
Returning to the context above, for A the initial object, the map
E(2) B0 C0! E(2) (B0 C0) (B0 C0) ! B0q C0
10 MICHAEL A. MANDELL
induces a map from the differential torsion product Tor*(B; C) to the E1 torsi*
*on
product E1 Tor*(B; C). In Section 14, we prove the following theorem.
Theorem 3.4. The map Tor*(B; C) ! E1 Tor*(B; C) is an isomorphism.
We use the previous theorem to construct a spectral sequence for the calculat*
*ion
of E1 Tor*A(B; C) for general A. For this, we need the bar construction in the *
*cate
gory of Ealgebras. Given Ealgebra maps A ! B and A ! C, the bar construction
fio(B; A; C) is the simplicial Ealgebra that is given in simplicial degree n by
fin(B; A; C) = B q A_q_._.q.Az____"qC:
n factors
Regarding B qA C as a constant simplicial Ealgebra, the map B q C ! B qA C
induces a map of simplicial Ealgebras fio(B; A; C) ! B qA C and therefore a map
of differential graded modules on their normalizations, N(fio(B; A; C)) ! B qA *
*C.
In fact the normalization of a simplicial Ealgebra is naturally an Ealgebra v*
*ia the
shuffle map [18, p. 51], and this is actually a map of Ealgebras, but we do no*
*t need
this fact here. The fact we do need is given in the following theorem, proved *
*in
Section 14.
Theorem 3.5. Let A, B be cofibrant Ealgebras, A ! B a map of Ealgebras and
A ! C a cofibration of Ealgebras. Then the canonical map
N(fi(B; A; C)) ! B qA C
is a quasiisomorphism.
Since Fp is a field, the following is an immediate consequence of the previous
theorem and Theorem 3.4. Although less immediate it still holds for E1 operads
over an arbitrary commutative ground ring.
Corollary 3.6.There is a left halfplane cohomological spectral sequence with
Ep;q2= Torp;qH*A(H*B; H*C);
converging strongly to E1 Torp+qA(B; C).
4. Construction of the Functor U
We construct the functor U whose restriction provides the inverse equivalence
of the Main Theorem. As mentioned in the introduction, we construct U as the
derived functor of a functor U from the category of Ealgebras to the category
of simplicial sets, adjoint to the cochain functor. We begin by reinterpreting *
*the
cochain functor as a limit.
Consider the cosimplicial simplicial set = [ .] given by the standard sim
plexes. Then C*[ .] is a simplicial Ealgebra. For an arbitrary set S, write
P (S; C*[n]) for the product of copies of C*[n] indexed on S. Then for a simpli
cial set X, P (X; C*[ .]) is a cosimplicial simplicial Ealgebra. Write M(X; C*)
for the end, the equalizer in the category of Ealgebras of the diagram
Q _____// Q *
P (Xn; C*[n])_____// P (Xm ; C [n]):
n f :m!n
f in op
By construction M(X; C*) is an Ealgebra, contravariantly functorial in the sim
plicial set X.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 11
Proposition 4.1.The cochain functor C* is canonically naturally isomorphic to
M(; C*) as a functor from simplicial sets to Ealgebras.
Proof.For each element of Xn, there is a canonical map [n] ! Xn, and the
collection of all such maps induces a map of Ealgebras
Q *
C*X ! P (Xn; C [n]):
n
By naturality, this map factors through the equalizer to induce a map of Ealge*
*bras
C*X ! M(X; C*). The underlying differential graded module of an equalizer of
Ealgebras is the equalizer of the underlying differential graded modules. It f*
*ollows
that the induced map C*X ! M(X; C*) is an isomorphism of the underlying *
*__
differential graded modules and therefore an isomorphism of Ealgebras. *
*__
The description of C* given by Proposition 4.1 makes it easy to recognize C*
as an adjoint. For an Ealgebra A, let UA be the simplicial set whose set of
nsimplices UnA is the mapping set E (A; C*[n]). Clearly UA is a contravari
ant functor of A. For a simplicial set X, the set of simplicial maps from X to
UA, opSet(X; UA) is by definition the end of the cosimplicial simplicial set
Setmn(X; UA) = Set(Xm ; UnA) that in cosimplicial degree m and simplicial degree
n consists of the set of maps of sets from Xm to UnA. Consider the cosimplicial
simplicial bijection
Set(Xm ; UnA) = Set(Xm ; E (A; C*[n]))
~=Q E (A; C*[n]) ~=E (A; Q C*[n]) = E (A; P (Xm ; C*[n])):
Xm Xm
Passing to ends gives a bijection
opSet(X; UA) ~=E (A; C*X);
natural in A and X. Thus, we have proved the following proposition.
Proposition 4.2.The functors U and C* are contravariant right adjoints between
the category of simplicial sets and the category of Ealgebras.
We now use the results of Section 2 on adjoint functors. Since we stated Theo
rems 2.14 and 2.15 in terms of covariant functors, we apply them to U, C* viewe*
*d as
an adjoint pair between the category of Ealgebras and the opposite to the cate*
*gory
of simplicial sets. As such, U is the left adjoint. Taking the closed model cat*
*egory
structure on the opposite category of simplicial as the one opposite to the sta*
*ndard
one [25] on the category of simplicial sets, the "fibrations" are the maps oppo*
*site
to monomorphisms and the "weak equivalences" are the maps opposite to weak
equivalences. It follows that the functor C* converts "fibrations" to surjectio*
*ns and
"weak equivalences" to quasiisomorphisms. It then follows from Theorems 2.14
and 2.15 that the left derived functor of U :E ! (opSet)op exists and is adjoint
to the right derived functor of C*: (opSet)op ! E . When we regard U as a con
travariant functor, this derived functor is the right derived functor, and we o*
*btain
the following proposition.
Proposition 4.3.The (right) derived functor U of U exists and gives an adjunc
tion hE (A; C*X) ~=H o(X; UA).
Applying Theorem 2.15 again, we obtain the following proposition, which is
needed in the proofs of Theorems 1.1 and 1.2 in the next section.
12 MICHAEL A. MANDELL
Proposition 4.4.The functor U converts cofibrations of Ealgebras to Kan fibra
tions of simplicial sets.
According to Theorem 2.14, the derived functor U is constructed by first ap
proximating an arbitrary Ealgebra with a cofibrant Ealgebra and then applying
U. This gives us the following standard observation.
Proposition 4.5.Let X be a simplicial set and A ! C*X a quasiisomorphism,
where A is a cofibrant Ealgebra. The unit of the derived adjunction X ! UC*X
is represented by the map X ! UA.
Instead of using the standard model structure on the category of simplicial s*
*ets,
we can use the "H*(; Fp)local" model structure constructed in [1]. In this st*
*ruc
ture, the cofibrations remain the monomorphisms but the weak equivalences are
the Fphomology equivalences. Since the functor C* has the stronger property of
converting Fphomology isomorphisms to quasiisomorphisms, the derived adjunc
tion factors as an adjunction between the homotopy category of Ealgebras and t*
*he
padic homotopy category. Although we do not need it in the remainder of our
work, we see that the functor U has the following strong H*(; Fp)local homoto*
*py
properties.
Proposition 4.6.The functor U converts Ealgebra cofibrations to H*(; Fp)local
fibrations. For a cofibrant Ealgebra A, UA is an H*(; Fp)local simplicial se*
*t.
5. The Fibration Theorems
In this section, we prove Theorems 1.1 and 1.2 that allow us to construct res*
*olv
able simplicial sets out of other resolvable simplicial sets. The proofs procee*
*d by
choosing cofibrant Ealgebra approximations and applying Propositions 4.4 and 4*
*.5
of the previous section.
Proof of Theorem 1.1.By Proposition 2.6, a map of Ealgebras can be factored as*
* a
cofibration followed by an acyclic fibration. Applying this to the Ealgebras C*
**Xn,
we can construct the following commutative diagram of Ealgebras.
Fp //____//A0//_____//_A1//__//././.__//An//____//. . .
  
~  ~ ~ 
fflfflfflfflfflfflfflffl fflfflfflffl
C*X0 ____//_C*X1____//._._._//C*Xn_____//. . .
Let A = ColimAn. From the universal property, we obtain a map A ! C*X. The
assumption that H*X = ColimH*Xn then implies that the map A ! C*X is a
quasiisomorphism.
Applying the functor U, we see that UA is the limit of UAn. We have the
following commutative diagram.
. ._.__//_//_Xn__////_._././_//_X1_////_X0
~  ~  ~
fflffl fflffl fflffl
. ._.__//_//_UAn_//_//_._././_//_UA1//_//_UA0
The bottom row is a tower of Kan fibrations by Proposition 4.4 and the vertical
maps are weak equivalences by Proposition 4.5 and the assumption that the Xn are
resolvable. It follows that the map of the limits X ! UA is a weak equivalence,_
and we conclude that A is resolvable. __
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 13
Remark 5.1.The argument above actually proves a more general result than stated
in Theorem 1.1. Let X1 = LimXn for a tower of Kan fibrations of Kan complexes.
Assume that each Xn is resolvable and that X1 is nonempty. If X ! X1 induces
an isomorphism H*X ! ColimH*Xn, then the argument above factors the unit
of the derived adjunction X ! UC*X through a natural isomorphism in the ho
motopy category X1 ! UC*X. If we assume the Main Theorem for a moment,
then when X is connected and of finite ptype, we can applying this observation
to the BousfieldKan pcompletion tower RnX for R = Fp. We conclude that for
any connected X of finite ptype, the unit of the derived adjunction X ! UC*X is
naturally isomorphic in the homotopy category to the BousfieldKan pcompletion
map X ! R1 X.
The proof of Theorem 1.2 is similar, but needs in addition the following resu*
*lt
that relates the E1 torsion product to the usual differential torsion product a*
*nd is
proved at the end of this section.
Lemma 5.2. Let X, Y , and Z be as in Theorem 1.2. The E1 torsion product
E1 Tor*C*Z(C*X; C*Y ) is canonically isomorphic to the usual differential torsi*
*on
product Tor*C*Z(C*X; C*Y ). Under this isomorphism, the composite
Tor*C*Z(C*X; C*Y ) ! H*(C*X qC*Z C*Y ) ! H*(C*(X xZ Y )) = H*(X xZ Y )
is the EilenbergMoore map.
Proof of Theorem 1.2.Using Proposition 2.6, choose cell Ealgebras A, B, C, qua*
*si
isomorphisms A ! C*Z, B ! C*X, C ! C*Y , and relative cell inclusions A ! B,
A ! C such that the following diagram commutes.
B oo_____oAo//_____//__C
  
~  ~  ~
fflfflfflfflfflfflfflfflfflfflfflffl
C*X oo___C*Z _____//C*Y
Let D = B qA C and consider the map D ! C*(X xZ Y ). By Lemma 5.2
and wellknown results on the EilenbergMoore map (e.g. [27, 3.2]), the map
D ! C*(X xZ Y ) is a quasiisomorphism. It follows that the unit of the derived
adjunction is represented for X xZ Y as the map X xZ Y ! UD. We have the
following commutative diagram.
X xZ YJ ______________//ZC
 JJJ  CCC
 JJJ  CC
 JJ  CC
 $$ ___________/!!/_
 UD  UC
   
   
fflfflfflffl fflfflfflffl
Y _______________//_JJXC 
JJJ  CCC 
JJJ  CC 
JJ%%fflfflfflfflC!fflfflfflffl!C
UB ____________//_UA
The assumption that X, Y , and Z are resolvable implies that all four maps betw*
*een
the top and bottom squares are weak equivalences, and we conclude that X xZ_Y
is resolvable. __
14 MICHAEL A. MANDELL
The proof of Lemma 5.2 consists of a comparison of the bar construction in
the category of Ealgebras with the cochain complex of the cobar construction of
simplicial sets. Recall that for maps of simplicial sets X ! Z and Y ! Z, the
cobar construction Cobaro(X; Z; Y ) is the cosimplicial simplicial set that is *
*given
in cosimplicial degree n by
Cobarn(X; Z; Y ) = X x Z_x_._.x.Zz____"xY
n factors
with face maps induced by diagonal maps and degeneracies by projections. The
cochain complex C*(Cobaro(X; Z; Y )) is then a simplicial Ealgebra. The nor
malization N(C*(Cobaro(X; Z; Y ))) is a differential graded Fpmodule; there is*
* a
canonical map from the usual differential torsion product to the homology
Tor*C*Z(C*X; C*Y ) ! H*(N(C*(Cobaro(X; Z; Y ))));
which is an isomorphism when X, Y , and Z are of finite ptype. On the other ha*
*nd,
considering X xZ Y as a cosimplicial simplicial set constant in the cosimplici*
*al
direction, the inclusion X xZ Y ! X x Y induces a map of cosimplicial simplicial
sets X xZ Y ! Cobaro(X; Z; Y ) and therefore a map of differential graded Fp
modules
N(C*(Cobaro(X; Z; Y ))) ! C*(X xZ Y ):
The composite map
Tor*C*Z(C*X; C*Y ) ! H*(N(C*(Cobaro(X; Z; Y )))) ! H*(X xZ Y )
is by definition the EilenbergMoore map; see for example [27].
Proof of Lemma 5.2.Let A, B, and C be as in the proof of Theorem 1.2 above.
The various projection maps of X x (Z x . .x.Z) x Y induce a map
B q (A q . .q.A) q C ! C*X q (C*Z q . .q.C*Z) q C*Y
! C*(X x (Z x . .x.Z) x Y ):
Theorem 3.4 and the K"unneth theorem imply that the composite above is a quasi
isomorphism. We obtain a degreewise quasiisomorphism of simplicial Ealgebras
fio(B; A; C)) ! C*(Cobaro(X; Z; Y ))
and therefore a quasiisomorphism of differential graded Fpmodules
N(fio(B; A; C)) ! N(C*(Cobaro(X; Z; Y )))
that makes the following diagram commute.
N(fio(B; A; C))~__//N(C*(Cobaro(X; Z; Y )))
~  
fflffl fflffl
B qA C _____________//C*(X xZ Y )
By Theorem 3.5, the left vertical arrow is a quasiisomorphism. By definition,
H*(B qA C) is the E1 torsion product E1 Tor*C*Z(C*X; C*Y ) and*the bottom
horizontal map induces on homology the canonical map E1 TorC Z(C*X; C*Y_)_!
C*X qC*Z C*Y . The lemma now follows. __
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 15
6. A Model for C*K(Z=pZ ; n)
In this section, we prove Theorem 1.3 that K(Z=pZ ; n) and K(Z^p; n) are re
solvable for n 1. We prove the resolvability of K(Z=pZ ; n) by constructing an
explicit cell Ealgebra model of C*K(Z=pZ ; n) that lets us analyze the unit of*
* the
derived adjunction. The case of Z^pfollows easily from the case of Z=pZ and the
work of the previous section.
The construction of our cell model requires the use of the generalized Steenr*
*od
operations for E1 algebras [18, xI.7], [22]. The theory of [22] gives Fplinea*
*r (but
not Fplinear) operations on the homology of an Ealgebra. In this section, we
only need the operation P 0. This operation preserves degree and performs the p*
*th
power operation on elements in degree zero. Using this fact, naturality, and th*
*e fact
that the operations commute with "suspension" [22, 3.3], the following observat*
*ion
can be proved by the argument of [22, 8.1].
Proposition 6.1.For any simplicial set X, the operation P 0on H*X induced
by the Ealgebra structure is the identity on elements of H*X in the image of
H*(X; Fp).
In Section 11, we describe all of the E1 algebra Steenrod operations on H*X in
terms of the usual Steenrod operations on H*(X; Fp).
For n 1, let Kn be a model for K(Z=pZ ; n) such that the set of nsimplices
of Kn is Z=pZ , e.g. the "minimal" model [21, x23]. Then we have a fundamental
cycle kn of CnKn which represents the cohomology class in HnKn that is the
image of the fundamental cohomology class of Hn(K(Z=pZ ; n); Fp). Write Fp[n]
for the differential graded Fpmodule consisting of Fp in degree n and zero in *
*all
other degrees, and let Fp[n] ! C*Kn be the map of differential graded Fpmodules
that sends 1 2 Fp to kn. Let E denote the free functor from differential graded*
* Fp
modules to Ealgebras. We obtain an induced map of Ealgebras a: EFp [n] ! C*Kn
that sends the fundamental class in of EFp [n] to the fundamental class kn of C*
**Kn.
The operation P 0is not the identity on the fundamental homology class of
EFp [n]. We obtain our cell Ealgebra model of C*Kn, by forcing (1  P 0)[in] to
be zero as follows. Let pn be an element of EFp [n] that represents (1  P 0)[*
*in].
Since (1  P 0)[kn] is zero in H*Kn, a(pn) is a boundary in CnKn. Choose an
element qn of Cn1Kn such that dqn = a(pn). Write CFp [n] for the cone on Fp[n],
the differential graded Fpmodule that is Fp in dimensions n  1 and n and zero
in all other dimensions, with the differential Fp ! Fp the identity. We have a
canonical map qn :CFp [n] ! C*Kn sending the generators to qn and a(pn). We
have a canonical map Fp[n] ! CFp [n], and a map pn :Fp[n] ! EFp [n] that sends
the generator 1 to the element pn. The diagram of differential graded kmodules
on the left below then commutes.
Fp[n]______//CFp [n] EFp [n]____//ECFp [n]
pn qn pn  qn
fflffl fflffl fflffl fflffl
EFp [n]__a__//_C*Kn EFp [n]_a___//C*Kn
It follows that the diagram of Ealgebras on the right above commutes. Let Bn
be the Ealgebra obtained from the following pushout diagram in the category of
16 MICHAEL A. MANDELL
Ealgebras.
EFp [n]____//ECFp [n]
pn 
fflffl fflffl
EFp [n]______//_Bn
We therefore obtain a map ff: Bn ! C*Kn. We prove the following theorem in
Section 12.
Theorem 6.2. The map ff: Bn ! C*Kn is a quasiisomorphism.
Corollary 6.3.Kn is resolvable.
Proof.Applying U to the pushout diagram that defines Bn, we obtain the following
pullback diagram of simplicial sets.
UBn ______//_UECFp [n]
 
 
fflfflfflfflfflfflfflffl
UEFp [n]_Upn_//UEFp [n]
The vertical maps are Kan fibrations since the inclusion EFp [n] ! ECFp [n] is a
cofibration. The following two propositions, 6.4 and 6.5, then imply that UBn i*
*s a
K(Z=pZ ; n).
By Theorem 6.2, the unit of the derived adjunction Kn ! UC*Kn is represented
by the map Kn ! UBn. Since UBn is a K(Z=pZ ; n), to see that the is a weak
equivalence, we just need to check that the induced map on ssn is an isomorphis*
*m.
The p distinct homotopy classes of maps from Sn to Kn induce maps C*Kn ! C*Sn
that differ on homology. It follows that the composite maps Bn ! C*Sn differ on
homology and are therefore different maps in hE . We conclude from the adjuncti*
*on
isomorphism hE (Bn; C*Sn) ~=H o(Sn; UBn) that the map Kn ! UBn is injective __
on ssn, and is therefore an isomorphism on ssn. __
Proposition 6.4.UECFp [n] is contractible.
Proof.ECFp [n] is a cell Ealgebra and the map Fp ! ECFp [n] is a quasiisomorp*
*h_
ism, so the map UECFp [n] ! UFp = * is a weak equivalence of Kan complexes. _*
*_
Proposition 6.5.UEFp [n] is a K(Fp ; n) and the map Upn induces on ssn the map
1  , where denotes the Frobenius automorphism of Fp.
Proof.We have canonical isomorphisms
UEFp [n] = E (EFp [n]; C*) ~=M (Fp [n]; C*);
where M denotes the category of differential graded Fpmodules. Thus, UEFp [n]
is the simplicial set which in dimension m is the set of cocycles in Cn[m]. Thi*
*s is
the minimal K(Fp ; n) [21, 23.7ff].
The map of simplicial sets [n] ! [n]=@[n] induces a bijection
E (EFp [n]; C*[n]) ~=M (Fp [n]; C*[n]) ~=M (Fp [n]; C*([n]=@[n])):
On the other hand, since Cn1([n]=@[n]) = 0, we have a canonical identification
E (EFp [n]; C*([n]=@[n])) ~=M (Fp [n]; C*([n]=@[n])) ~=Hn([n]=@[n]):
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 17
By naturality, the map Hn([n]=@[n]) ! Hn([n]=@[n]) induced by pn must
be 1  P 0. Under the isomorphism
Hn([n]=@[n]) ~=Hn([n]=@[n]; Fp) Fp ~=Fp;
we can identify the operation 1  P 0as 1  by Proposition 6.1 and the Cartan_
formula [22, 2.7ff]. __
We complete the proof of Theorem 1.3 by deducing that K(Z^p; n) is resolvable
for n 1.
Proof of Theorem 1.3.We see by induction and Theorem 1.2 that K(Z=pm Z; n) is
resolvable for n 1 by considering the following fiber square
K(Z=pm Z; n)_____//_P K(Z=pZ ; n + 1)
 
 
fflfflfflffl fflfflfflffl
K(Z=pm1 Z; n)_____//_K(Z=pZ ; n + 1);
where P K(Z=pZ ; n + 1) is some contractible simplicial set with a Kan fibratio*
*n to
K(Z=pZ ; n + 1). Since K(Z^p; n) can be constructed as the limit of a tower of *
*Kan
fibrations
. .!.K(Z=pm Z; n) ! . .!.K(Z=pZ ; n);
and the natural map H*K(Z^p; n) ! ColimH*K(Z=pm Z; n) is an isomorphism, we __
conclude from Theorem 1.1 that K(Z^p; n) is resolvable. __
7.The Image Subcategory
The purpose of this section is to identify the Ealgebras that are quasiisom*
*orphic
to the cochain complexes of connected (pcomplete) nilpotent simplicial sets of
finite ptype. As mentioned in the introduction, the condition characterizing t*
*hese
Ealgebras is essentially the Ealgebra analogue of the existence of a finite p*
*type
principal Postnikov tower. Since the functors relating Ealgebras and simplici*
*al
sets are contravariant, towers of principal fibrations of simplicial sets corre*
*spond to
"complexes" of Ealgebras, formed by attaching "cells". We make this precise in
the following definitions.
Definition 7.1.An augmented Ealgebra is an Ealgebra B together with a map
of Ealgebras B ! Fp (the augmentation). A Bcell (CB; B) is an augmented
Ealgebra CB together with a cofibration of augmented Ealgebras B ! CB such
that the augmentation CB ! Fp is a quasiisomorphism. For a map of Ealgebras
f :B ! A, we say that A qB CB is formed by attaching a Bcell along f.
The cells we use are built out of the cell Ealgebras Bn of Section 6. Since
Fp = E0, the maps of differential graded Fpmodules Fp[n] ! 0 and CFp [n] ! 0
induce a map of Ealgebras Bn ! Fp that we take as an augmentation. Let
(CBn; Bn) be a Bncell.
Let B1;n= Bn. By the Main Theorem, we can choose a map fin :Bn+1 ! B1;n
that sends the fundamental class of Hn+1Bn+1 to the Bockstein class of Hn+1B1;n.
Let B2;n= B1;nqBn+1 CBn+1. Then UB2;nis a K(Z=p2Z; n) and it follows from
Lemma 5.2 that the map B2;n! C*K(Z=p2Z; n) is a quasiisomorphism. By the
Main Theorem, we can find a map fi2: Bn+1 ! B2;nthat sends the fundamental
class of Hn+1Bn+1 to the class in Hn+1B2;ncorresponding to the second Bockstein
18 MICHAEL A. MANDELL
in Hn+1K(Z=p2Z; n). Inductively, we can form Bm;n together with a cofibration
Bm1;n ! Bm;n and a quasiisomorphism Bm;n ! C*K(Z=pm Z; n) by attach
ing the Bn+1cell (CBn+1; Bn+1) along the map fim1 , and we can choose a map
fim :Bn+1 ! Bm;n that sends the fundamental class of Hn+1Bn+1 to the class in
Hn+1Bm;n corresponding to the mth Bockstein in Hn+1K(Z=pm Z; n).
Let B1;n = Colim Bm;n. The quasiisomorphisms Bm;n ! C*K(Z=pm Z; n)
induce a quasiisomorphism B1;n ! C*K(Z^p; n). Let B^n= B1;n, and choose a
B^ncell (CB^n; B^n). We can now define the "complexes" that we work with.
Definition 7.2.A B*complex is an Ealgebra A = ColimAj such that A0 = Fp
and for each j > 0 either Aj+1 = Aj or Aj+1 is formed from Aj by attaching a
Bmj;nj+1cell for some mj 1 or mj = 1, where {nj} is some unbounded non
decreasing sequence of positive numbers. A special B*complex is a B*complex in
which for each j, either mj = 1 and the Bnjcell is (CBnj; Bnj) or mj = 1 and
the B^njcell is (CB^nj; B^nj).
We allow the case Aj+1 = Aj in order to permit the possibility that A = Aj
for some j. The assumption that the nondecreasing sequence of positive integers
{nj} be unbounded is equivalent to the requirement that it repeat a given number
at most finitely many times. Thus, a B*complex is an Ealgebra formed from Fp
by inductively attaching Bm;n+1cells, for nondecreasing n, finitely many for *
*each
n 1. The analogy between B*complexes and Postnikov towers is made clear by
the following theorem and its proof.
Theorem 7.3. The following conditions on an Ealgebra A are equivalent.
(i)A is quasiisomorphic to C*X for some connected (pcomplete) nilpotent sim
plicial set of finite ptype.
(ii)A is quasiisomorphic to a B*complex.
(iii)A is quasiisomorphic to a special B*complex.
Thus, the homotopy category of connected pcomplete nilpotent spaces of finite *
*p
type is equivalent to the full subcategory of the homotopy category of Ealgebr*
*as
consisting of the special B*complexes.
Proof.We prove (i)=) (iii)=) (ii)=) (i).
Suppose A is quasiisomorphic to a connected nilpotent simplicial set of fini*
*te
ptype X; replacing X be its pcompletion if necessary, we can assume that X is
pcomplete. Then X has a principally refined Postnikov tower Xj whose fibers
are all K(Z=pZ ; n)'s and K(Z^p; n)'s with at most finitely many of each type f*
*or
each n. Lemma 5.2 allows us to approximate inductively C*Xj by the jth stage
of a special B*complex. In the colimit, we obtain a special B*complex and a
quasiisomorphism to C*X. This proves (i)=) (iii).
The implication (iii)=) (ii)is trivial.
For the implication (ii)=) (i), start with a B*complex A = ColimAj. By
Proposition 4.4, UA = Lim UAj is the limit of principal Kan fibrations of Kan
complexes. In fact, by the construction of the Bm;n's, UA = Lim UAj is a princi
pally refined Postnikov tower whose fibers are K(Z=pm Z; n)'s and K(Z^p; n)'s w*
*ith
only finitely many for each n. In particular, UA and each UAj are connected p
complete nilpotent simplicial sets of finite ptype. Clearly, Fp = A0 ! C*UA0
is a quasiisomorphism. Inductive application of Lemma 5.2 shows that the maps
Aj ! C*UAj are weak equivalences, and we conclude that the map A ! C*X is_
a quasiisomorphism. __
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 19
Remark 7.4.We can refine the argument above to see that an Ealgebra is equiva
lent to a finite stage special B*complex if and only if it is quasiisomorphic*
* to C*X
for some space X that has a finite stage finite type principally refined Postni*
*kov
tower. Likewise, an Ealgebra is equivalent to a finite stage special B*comple*
*x with
no B^n+1cells (for all n) if and only if it is equivalent to C*X for some spac*
*e X with
only finitely many nontrivial homotopy groups, all of which are finite pgroups*
*. If
we choose CBn to be a finite cell Ealgebra, such a B*complex is then also a f*
*inite
cell Ealgebra.
8.The Characterization Theorem
We defined B*complexes in the last section having in mind an analogy with the
definition of a principally refined Postnikov tower. We prove the Characterizat*
*ion
Theorem in this section having in mind an analogy with the construction of the
principal Postnikov tower of a simply connected space. Usually the main tool in*
* the
construction of a principal Postnikov tower is the Hurewicz theorem, of which we
have no analogue in the category of Ealgebras. Instead, we are forced to work *
*with
the EilenbergMoore spectral sequence of Corollary 3.6 and implicitly a Bockste*
*in
spectral sequence. To avoid repeating lengthy hypotheses, we use the following
terminology in this section and the next two.
Definition 8.1.We say that an Ealgebra A is 1connected if HnA = 0 for n < 0,
H0A = Fp, and H1A = 0. We say that A is finite type if for each n, HnA is a
finite dimensional Fpmodule. When A is finite type, we say that A is spacelike*
* if
for each n, HnA is generated as an Fpmodule by fixed points of P 0.
Definition 8.2.Let f :A ! B be a map of Ealgebras. We say that f is an
nequivalence if the induced map HiA ! HiB is an isomorphism for i < n and
an injection for i = n. We say that f is an napproximation if the induced map
HiA ! HiB is an isomorphism for i n.
Most of the work needed for the proof of the Characterization Theorem goes in*
*to
the following two lemmas. We prove these in the next two sections.
Lemma 8.3. Let A be a B*complex, let B be a 1connected finite type spacelike
Ealgebra, and let f :A ! B be an nequivalence of Ealgebras for some n 1.
Then f factors through an napproximation f0:A0 ! B such that A0 is formed
from A by attaching a finite number of B1;n+1cells.
Lemma 8.4. Let A be a B*complex, let B be a 1connected finite type spacelike
Ealgebra, and let f :A ! B be an napproximation of Ealgebras for some n 1.
If f is not an (n + 1)equivalence, then f factors through a map f0:A0! B such
that dim(kerHn+1f0) < dim(kerHn+1f), and A0 is formed from A by attaching a
single Bm;n+1cell for some m 1 or m = 1.
Proof of the Characterization Theorem.Let B be a 1connected finite type space
like Ealgebra. Then the map Fp ! B is a 2equivalence. Alternately applying
Lemma 8.3 and applying Lemma 8.4 (multiple times) inductively constructs a B*
complex A and a quasiisomorphism A ! B. Since in the construction of A, each
Bm;n+1cell attached has n 2, UA is 1connected. The Characterization Theorem_
then follows from the Main Theorem and Theorem 7.3. __
20 MICHAEL A. MANDELL
9.Cofiber Sequences and the Proof of Lemma 8.3
This section is devoted to the proof of Lemma 8.3. Thinking in terms of the
analogous lemma for spaces, we should be able to attach the B1;n+1cells in the
statement along the trivial map. The proof of Lemma 8.3 then reduces to finding
maps from Fp qB1;n+1CB1;n+1to B for B1;n+1cells (CB1;n+1; Bn+1). We find
these by working with cofiber sequences; the following definitions are standard.
Definition 9.1.Let A be a cofibrant augmented Ealgebra, and let IA be a cylin
der object for A. We define the cone of A to be the augmented Ealgebra CA =
IA qA Fp (via @1). We define the suspension of A to be the Ealgebra SA =
Fp qA CA (via @0). For any cofibrant Ealgebra B and any map of Ealgebras
f :A ! B, the cofiber of f is the Ealgebra Cf = B qA CA.
Choosing a diagonal lift in the diagram
@0q@1
A qfA___________//flfflIA5q@A5q_IA
_____________1 @0
@0+@1 ___________~_oe+oe___
fflffl_____________fflfflfflffl
IA ________oe_______//A
induces a map SA ! SA q SA and a map Cf ! Cf q SA. Just as in a closed
model category, these maps make SA is a cogroup object in hE and give Cf an
SA coaction in hE . In particular, for any Ealgebra D, hE (SA; D) is natural*
*ly
a group and hE (Cf; D) is naturally a hE (SA; D)set. It is not hard to see th*
*at
these structures are independent of the choice of lift used, and that S extends*
* to
a functor from hE to cogroup objects in hE . We have a canonical inclusion map
b: B ! Cf, and when in addition B is an augmented Ealgebra and f is a map of
augmented Ealgebras, we obtain a canonical collapse map c: Cf ! SA. Although
usually stated in the "pointed" context, the usual arguments (e.g. [25, x3]) ap*
*ply
in this context to show that the maps Sqf, Sqb, and Sqc induce a long "exact"
sequence of mapping set functors for q 0. We state this only in the case we ne*
*ed,
avoiding the complications of the q = 0 case.
Proposition 9.2.Let A and B be cofibrant augmented Ealgebras, and let f :A ! B
be a map of augmented Ealgebras. For an arbitrary Ealgebra D, the sequence
hE (S2B; D) S2f*!hE (S2A; D) Sc*!hE (SCf; D) Sb*!hE (SB; D) Sf*!hE (SA*
*; D)
is an exact sequence of groups.
For a free Ealgebra EX, the diagonal map X ! X X induces a comultiplica
tion EX ! E(X X) ~=EX q EX, making EX into an abelian cogroup object in
E . It is not hard to see that suspension commutes with the free functor. We ne*
*ed
this observation only in the simplest case, which we state as the first part of*
* the
following proposition.
Proposition 9.3.For each n, there is a canonical isomorphism of EFp [n] and
SEFp [n + 1] as cogroup objects in hE . The induced natural transformation
oe :Hn+1Fp [m + 1] = hE (Fp [n + 1]; Fp[m + 1]) !
hE (SFp [n + 1]; SFp [m + 1]) ~=hE (Fp [n]; Fp[m]) = HnFp [m]
commutes with the homology operations P sfor all s.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 21
Proof.Let *[1] = C*[1] denote the standard 1simplex differential graded Fp
module. Then E(Fp [n + 1] *[1]) is a cylinder object for EFp [n + 1] and there
are canonical isomorphisms
EFp [n] ~=E(Fp [n + 1] *[1]=(Fp Fp))
~=FpqEFp[n+1]E(Fp [n + 1] *[1]) qEFp[n+1]Fp= SEFp [n + 1]:
It is easy to check that this is an isomorphism of abelian cogroups in E . The*
* fact
that the operations P scommute with oe for A = EFp [n] follows from looking at *
*the
sequence
EFp [m] ! ECFp [m + 1] ! EFp [m + 1];
and applying [22, 3.3]. ___
The following proposition is an easy consequence of the previous proposition *
*or
of the Main Theorem and Theorem 7.3.
Proposition 9.4.For n > 0, there is an isomorphism (in hE ), Bn ' SBn+1.
The augmented Ealgebra Bn+1 is the cofiber of the map pn+1: EFp [n + 1] !
EFp [n + 1]. Proposition 9.3 and the naturality of the operation 1  P 0identif*
*ies
the exact sequence of Proposition 9.2 for Bn+1 as the sequence
0 1P0
Hn1D 1P!Hn1D ! hE (SBn+1; D) ! HnD ! HnD:
This allows us to identify the mapping group hE (SBn+1; D) when we understand
the operation P 0on H*D. In particular, when 1  P 0is surjective on Hn1D, we
obtain an isomorphism (of groups) between hE (SBn+1; D) and the kernel of 1P 0
on HnD. Although we use this principally for SBn+1, the identification is most
naturally stated via Proposition 9.4 in terms of Bn.
Proposition 9.5.Let n > 0 and let D be an Ealgebra. If the operation 1  P 0is
surjective on Hn1D, then the association of a map Bn ! D to the element of HnD
given by image of the fundamental class induces a bijection between hE (Bn; D) *
*and
the fixed points of P 0in HnD.
Proof of Lemma 8.3.Choose a basis for the fixed points of P 0in H*
*nA
and expand this to a basis of the fixed points of P *
*0in HnB.
By the previous two propositions, we can find maps fj:SBn+1 ! B such that the
composite Bn ! SBn+1 ! B sends the fundamental class of Bn to bj. Then
A0= A q SBn+1 q . .q.SBn+1
is formed from A by attaching a finite number of B1;n+1cells and the map
f0 = f + f1 + . .+.fr:A q SBn+1 q . .q.SBn+1 ! B
is easily seen by Theorem 3.4 to be an napproximation. __*
*_
10.Proof of Lemma 8.4
The definition of B*complex in Section 7 and the statement of Lemma 8.4 leave
us complete freedom in choosing the cells (CBm;n+1; Bm;n+1). We take advantage
of this freedom here in the proof of Lemma 8.4. In particular, we choose specif*
*ic cells
(Cm;n+1; Bm;n+1) which admit cofibrations Cm;n+1 ! Cm+1;n+1 under the maps
Bm;n+1 ! Bm+1;n+1 such that for C1;n+1 = Colimm Cm;n+1, (C1;n+1; B1;n+1)
22 MICHAEL A. MANDELL
is a B1;n+1cell. Regarding these cells, we have the following lemma, the proof*
* of
which occupies most of this section.
Lemma 10.1. Let f :A ! B be as in Lemma 8.4, and let x 2 ker(Hn+1f) be a
nonzero fixed point of P 0. Let g :Bm;n+1 ! A be a map that sends the fundamen*
*tal
class of Hn+1Bm;n+1 to x for some 1 m < 1, and let
fg: Ag = A qBm;n+1Cm;n+1 ! B
be some map extending f. If dim(kerHn+1fg) = dim(kerHn+1f), then g extends
to a map h: Bm+1;n+1 ! A and fg extends to a map
fh: Ah = A qBm+1;n+1Cm+1;n+1 ! B:
Proof of Lemma 8.4 from Lemma 10.1.Choose a map g1: B1;n+1! A sending the
fundamental class to x. By Proposition 9.5, the composite f O g1: B1;n+1! B is
homotopic to the trivial map (the augmentation of B1;n+1composed with the unit
of B). It follows that f O g1 extends to a map
A qB1;n+1IB1;n+1qB1;n+1Fp! B
for a cylinder object IB1;n+1. By choosing a diagonal lift in the diagram
B1;n+1_____//fflfflIB1;n+1qB1;n+1Fp
__77____
 _________~_
 _________ 
fflffl_____ fflfflfflffl
C1;n+1____________//Fp
and composing with the map above, we obtain a map
fg1:Ag1= A qB1;n+1C1;n+1! B
extending f. Inductively, for as long as possible, choose maps gm+1 :Bm+1;n+1 !*
* A
and fgm+1:Agm+1 ! B extending the maps gm and fgm. If gm ; fgm cannot be
extended to some gm+1 ; fm+1 , then A0 = Agm, f0 = fgm satifies the conclusion
of Lemma 8.3 by Lemma 10.1. Otherwise, let g = Colimgm , let A0= A qB1;n+1
C1;n+1, and let f0 = Colimfgm,
f0:A0= A qB1;n+1C1;n+1 = A qB1;n+1Colim Cm;n+1 ! B:
We have assumed that the map B1;n+1 ! C1;n+1 is a cofibration, and so we can
use the EilenbergMoore spectral sequence of Corollary 3.6 to calculate the eff*
*ect
on homology of the inclusion of A in A0: It is an isomorphism on Hi for i < n +*
* 1
and is the quotient by the submodule generated by x on Hn+1. It follows that the
image of Hn+1A0in Hn+1B coincides with the image of Hn+1A, but the dimension
of Hn+1A0 is one less than the dimension of Hn+1A, and so the dimension of __
kerHn+1f0 is one less than the dimension of kerHn+1f. __
Recall that the augmented Ealgebras Bm;n+1 for m > 1 are constructed induc
tively by attaching a Bn+2cell along a map fim1 :Bn+2 ! Bm1;n+1 representing
the (m  1)st Bockstein. We did not specify how to choose the Bn+2cell in Sec
tion 7; we now assume that CBn+2 is a cone IBn+2 qBn+2 Fp for some cylinder
object IBn+2. Let S = CBn+2 qBn+2 CBn+2. For later convenience, choose and
fix a quasiisomorphism q :Bn+1 ! SBn+2 ! S. Let IS be a cylinder object for
S, and let CS be the cone IS qS Fp.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 23
We choose the cell (C1;n+1; B1;n+1) arbitrarily and for m > 1 we choose the c*
*ells
(Cm;n+1; Bm;n+1) inductively as follows. Choose a map CBn+2 ! Cm;n+1 so that
the restriction Bn+2 ! Cm;n+1 factors through the map fim :Bn+2 ! Bm;n+1, i.e.
find a diagonal lift in the following diagram.
Bn+2fflfflfim//_Bm;n+1////_Cm;n+144_____
_______
 _____________
 ____________ fflffl~
fflffl__________ fflffl
CBn+2 ___________________//Fp
Consider the map
fln :S = CBn+2 qBn+2 CBn+2 ! CBn+2 qBn+2 Cm;n+1
and let Cm+1;n+1 = Cfln be the cofiber. Since fln is a quasiisomorphism by The*
*o
rem 3.2 and an augmented map, Cm+1;n+1 is augmented and its augmentation is a
quasiisomorphism. Note that Bm+1;n+1qBm;n+1Cm;n+1 = CBn+2 qBn+2 Cm;n+1,
and so we obtain cofibrations Bm+1;n+1 ! Cm+1;n+1 and Cm;n+1 ! Cm+1;n+1
under Bm;n+1. We let C1;n+1 = ColimCm;n+1 as required. Since by construction
the maps Bm+1;n+1qBm;n+1Cm;n+1 ! Cm+1;n+1 are cofibrations for all m, we have
that the map B1;n+1 = ColimBm;n+1 ! C1;m+1 is a cofibration.
Proof of Lemma 10.1.Let g, Ag, and fg be as in the statement and suppose that
dim(kerHn+1fg) = dim(kerHn+1f). Looking at the EilenbergMoore spectral
sequence of Corollary 3.6 that calculates the homology of Ag, we see that the
composite map g O fim :Bn+2 ! A must send the fundamental class to zero, and
the image of Hn+1fg must be the same as the image of Hn+1f, since otherwise
we would have dim(kerHn+1fg) = dim(kerHn+1f)  1. Since g O fim sends the
fundamental class to zero, it follows from Proposition 9.5 that we can extend g*
* Ofim
to a map b: CBn+2 ! A. Using the map CBn+2 ! Cm;n+1 in the construction
of Cm;n+1, we obtain a map a: S ! Ag; let y be the image in Hn+1Ag of the
fundamental class of Hn+1Bn+1 under the map a O q :Bn+1 ! Ag.
We can change the choice of map b by "adding" a map c: SBn+2 ! A via the map
CBn+2 ! CBn+2qSBn+2 under Bn+2. Doing so changes y by adding the image of
the fundamental class in Hn+1Ag of c composed with A ! Ag. In particular, since
the image of Hn+1A in Hn+1B coincides with the image of Hn+1Ag in Hn+1B, we
can choose b so that y is in the kernel of Hn+1fg. Let h: Bm+1;n+1 ! A be the
map induced by g together with such a choice of b. Then the composite map
S ! Bm+1;n+1qBm;n+1Cm;n+1 ! Ag ! B
sends the fundamental class of Hn+1Bn+1 ~= Hn+1S to zero, and so this map
extends to a map CS ! B. This specifies a map
fh: A qBm+1;n+1Cm+1;n+1 ~=(Ag qBm;n+1Bm+1;n+1) qS CS ! B
that extends fg. ___
11.The Algebra of Generalized Steenrod Operations
The key to the proof of Theorem 6.2 is a study of the algebra of all generali*
*zed
Steenrod operations of [22]. Precisely, let B be the free associative Fpalgeb*
*ra
generated by the P sand (if p > 2) the fiP s[22, 2.2,x5] for all s 2 Z modulo t*
*he
twosided ideal consisting of those operations that are zero on all "Adem objec*
*ts"
24 MICHAEL A. MANDELL
[22, 4.1] of "C (p; 1)" of [22, 2.1]. The Adem objects of C (p; 1) include all *
*E1
algebras over any E1 koperad for any commutative Fpalgebra k. In this sectio*
*n,
we prove Theorem 1.4 and provide the main results needed in the next section
to prove Theorem 6.2. We use the standard arguments effective in studying the
Steenrod and DyerLashoff algebras to analyze the structure of B.
Definition 11.1.We define length, admissibility, and excess as follows
(i)p = 2: Consider sequences I = (s1; : :;:sk). The sequence I determines the
operation P I= P s1. .P.sk. We define the length of I to be k. Say that I *
*is
admissible if sj 2sj+1 for 1 j < k. We define the excess of I by
k1X Xk
e(I) = sk + (sj 2sj+1) = s1  sj
j=1 j=2
(ii)p > 2: Consider sequences I = (ffl1; s1; : :;:fflk; sk) such that ffli is *
*0 or 1. The
sequence I determines the operation P I= fiffl1P s1. .f.ifflkP sk, where f*
*i0P s
means P sand fi1P smeans fiP s. We define the length of I to be k. Say that
I is admissible if sj psj+1+ fflj+1. We define the excess of I by
k1X Xk
e(I) = 2sk + ffl1 + (2sj 2psj+1 fflj+1) = 2s1 + ffl1  (2sj(p  1) + f*
*flj)
j=1 j=2
In either case, by convention, the empty sequence determines the identity opera*
*tion,
has length zero, is admissible, and has excess 1. If I and J are sequences, we
denote by (I; J) their concatenation.
Proposition 11.2.The set {P I I is admissible} is a basis of the underlying Fp
module of B.
Proof.It follows from the Adem relations [22, 4.7] that the set generates B as *
*a Fp
module. Linear independence follows by examination of the action on H*(GFp [n])
as n gets large, where G denotes the free Galgebra functor for some E1 Fp ope*
*rad_
G. This follows for example from [23, 2.2 or 2.6]. _*
*_
Proposition 11.3.If s > 0 then P s(P 0)s = 0 and (if p > 2) fiP s(P 0)s = 0.
Proof.Here P s(P 0)s and fiP s(P 0)s are meant to denote P sor fiP scomposed
with s factors of P 0. The Adem relations [22, 4.7] for fifflP sP 0when s > 0 *
*are
given by
1X (p  1)i  1
fifflP sP 0= (1)si fifflP (si)P i;
i=1 s  i  1
where we understand ffl = 0 when p = 2, and we understand the binomial coeffici*
*ent
n
k = 0 when k < 0 or k > n. The binomial coefficient in the expression above
therefore can be nonzero only when s=p i s  1. Then P 1P 0= 0 and
fiP 1P 0= 0 since for these the coefficients are zero for all values of i. Ass*
*ume by
induction that P t(P 0)t= 0 for all t such that 1 t < s; we see that P sP 0a*
*nd
fiP sP 0are both in the left ideal generated by {P t 1 t < s} and hence by
the inductive hypothesis are annihilated by (P 0)s1; therefore, P s(P 0)s_= 0*
* and
fiP s(P 0)s = 0. __
We can now prove the first half of Theorem 1.4.
Proposition 11.4.The left ideal of B generated by (1  P 0) is a twosided idea*
*l.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 25
Proof.By Proposition 11.2, it suffices to show that for every admissible sequen*
*ce
I, (1  P 0)P Iis an element of the left ideal generated by (1  P 0). Let I =
(ffl1; s1; : :;:fflk; sk) be an admissible sequence (where if p = 2 each fflj =*
* 0 and we
think of this sequence as (s1; : :;:sk)). If sk < 0 then by the previous propos*
*ition
P I= P I(1  (P 0)sk) = P I(1 + P 0+ . .+.(P 0)sk1)(1  P 0)
is in the ideal and hence (1P 0)P Iis as well. We can therefore assume that sk*
* 0,
and it follows from admissibility that sj 0 for all j. We proceed by induction*
* on
k, the length of I.
The statement is trivial for k = 0 (the empty sequence); now assume by induct*
*ion
that the statement holds for all sequences J of length less than k. We can writ*
*e I
as the concatenation ((ffl; s); J) for some sequence J of length k  1. If s = *
*0, the
Adem relation for P 0fiP 0is P 0fiP 0= fiP 0P 0, and we see that
(1  P 0)P I= (1  P 0)fifflP 0P J= fifflP 0(1  P 0)P J
is in the ideal by induction. For s > 0, the Adem relation for P 0P stakes the *
*form
X (p  1)(s  i)  1
P 0P s= (1)i (p  1)s + i  1P siP i:
When i > 0 the binomial coefficient is zero, when i = 0 we get the term P sP 0,
and when i < 0 we get terms of the form binomial coefficient times P siP ithat
we know from the work above are in the ideal; therefore, we can write P 0P s=
P sP 0+ ff(1  P 0) for some ff. An entirely similar argument shows that P 0fiP*
* scan
also be written P 0fiP s= fiP sP 0+ ff(1  P 0) for some ff. It follows that
(1  P 0)P I= (1  P 0)fifflP sP J= (fifflP s+ ff)(1  P 0)P J
is in the ideal by induction, and this completes the argument. *
*___
For the other half of Theorem 1.4, we need a canonical map from B to the
Steenrod algebra A. It can be shown [22, 10.5] that the Steenrod operations on *
*the
cohomology of a simplicial set arise from the action of B from a C (p; 1) struc*
*ture
on the cochains with coefficients in Fp. However, it is important for our purpo*
*ses
to relate the action of B obtained from the Ealgebra structure to the Steenrod
algebra. The previous proposition implies that if x is an element of a left B
module that is fixed by P 0, then the submodule Bx generated by x is fixed by
P 0. It follows from this observation and Proposition 6.1 that for any simplici*
*al set
X, the Fpsubmodule H*(X; Fp) of H*X is a Bsubmodule. It then follows from
the axioms that uniquely identify the Steenrod operations that the action of P *
*son
H*(X; Fp) coincide with the Steenrod operations of the same name. Furthermore,
by looking at C*Kn, it is possible to identify fiP sas the composite of the ope*
*ration
P sand the Bockstein. Thus, we understand the canonical map B ! A as follows.
Proposition 11.5.Let k be a commutative Fpalgebra and let G be an E1 operad
of differential graded kalgebras. For any Galgebra structure on C*(X; k) that*
* is
natural in the simplicial set X, the operations P sand (for p > 2) fiP sact on *
*an
element of H*(X; Fp) H*(X; k) by the Steenrod operations of the same name.
Remark 11.6.The previous proposition and the Cartan formula [22, 2.2] allow the
identification the operations on H*X in terms of the Steenrod operations. When
X is of finite ptype, H*X ~=H*(X; Fp) Fp Fp, and so every element of HnX can
26 MICHAEL A. MANDELL
be written as a linear combination a1x1+ . .+.am xm for some elements x1; : :;:*
*xm
in H*(X; Fp) and a1; : :;:am in Fp. Then
fifflP s(a1x1 + . .+.am xm ) = (a1)fifflP sx1 + . .+.(am )fifflP sxm ;
where denotes the Frobenius automorphism of Fp. In general H*X is the limit
of H*Xffwhere Xffranges over the finite subcomplexes of X.
Proof of Theorem 1.4.The map B ! A is clearly surjective. Since the relation
P 0= 1 holds in A, the map B ! A factors through the ring B=(1  P 0) and
certainly remains surjective. To see that it is injective, note that Propositio*
*ns 11.2
and 11.3 imply that B=(1  P 0) is generated as an Fpmodule by those P Ifor
admissible sequences I = (ffl1; s1; : :;:fflk; sk) such that sj > 0 for each j;*
* the image
of these elements in A form an Fpmodule basis, and in particular are linearly_
independent. __
12.Unstable Modules over B
In this section, we prove Theorem 6.2. The proof is based on a comparison of
free unstable modules over A with free unstable modules over B.
Definition 12.1.A module M over B is unstable if for every x 2 M, P Ix = 0 for
any I with excess e(I) greater than the degree of x.
Observe that a module over the Steenrod algebra is unstable if and only if it
is unstable as a module over B. Also observe that if M = H*A for an object
of C (p; 1), e.g. an E1 kalgebra A for a commutative Fpalgebra k, then M is
unstable [22, 5.(3)(4)].
We denote by Aunnand Bunnthe free unstable A and Bmodules on one generator
in degree n; we denote these generators as an and bn respectively. The following
proposition generalizes the standard basis theorem for Aunnand follows easily f*
*rom
Proposition 11.2.
Proposition 12.2.The set {P Ibn  I is admissible and e(I)} nis an Fpmodule
basis of Bunn.
We can identify the Bmodules H*EFp [n] in terms of free unstable Bmodules.
For this, we need the following terminology.
Definition 12.3.A restricted Fpmodule is a graded Fpmodule M together with
an additive endomorphism (the restriction) that multiplies degrees by p, i.e. t*
*akes
elements of degree n to elements of degree np. The enveloping algebra of M is t*
*he
free graded commutative Fpalgebra on M modulo the relation that the restriction
is the pth power operation.
An unstable Bmodule is naturally a restricted Fpmodule by neglect of struc
ture; its enveloping algebra inherits an unstable Bmodule structure via the Ca*
*rtan
formula.
Proposition 12.4.If G is an E1 Fpoperad, then H*GFp [n] is the enveloping
algebra of the free unstable Bmodule on one generator in degree n. H*EFp [n] is
the extended Fpalgebra on the enveloping algebra of Bunn.
Proof.The argument of [23, 2.6] applies to prove the first statement. The secon*
*d_
statement follows from the first. __
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 27
To use this in the proof of Theorem 6.2, we need to understand the map on
homology induced by the map EFp [n] ! EFp [n] in the construction of Bn. In the
following proposition, (1  P 0) denotes the map of Bmodules Bunn! Bunnthat
sends the generator to 1  P 0times the generator.
Proposition 12.5.For n 1, the sequence
0)
0 ! Bunn(1P!Bunn! Aunn! 0
is exact and split in the category of restricted Fpmodules.
Proof.The fact that Bunn! Aunnis onto is clear since it is a map of Bmodules a*
*nd
Aunnis generated as a Bmodule by the image of the generator of Bunn. Similarly,
exactness in the middle is clear from examination of the Fpmodule bases of B a*
*nd
A. Thus, it remains to show that the map Bunn! Bunnis injective and split in the
category of restricted Fpmodules.
We proceed by writing an explicit splitting f :Bunn! Bunnin the category of
restricted Fpmodules as follows. It suffices to specify f on P Ibn for each ad*
*missible
I = (ffl1; s1; : :;:fflk; sk) with e(I) n. If sk < 0, choose
f(P Ibn) = P I(1 + P 0+ (P 0)2 + . .).bn:
This is welldefined by Proposition 11.3. If fflk+ sk > 0 or if I is empty, the*
*n choose
f(P Ibn) to be zero. Let n(I) denote the largest number n such that the subsequ*
*ence
(fflkn+1; skn+1; : :;:fflk; sk) is all zeros; if fflk 6= 0 or sk 6= 0 then n(*
*I)=0. We have
chosen f(P I) when n(I) is zero; when n(I) > 0, writing I as the concatenation
(J; (0; 0)) where n(J) = n(I)  1, we inductively choose
f(P Ibn) = P Jbn + f(P Jbn):
It is immediate from the construction and the fact that pth power operations*
* do
not change the excess that f is a map of restricted Fpmodules. We need to veri*
*fy
that the composite of f and the map (1  P 0) is the identity. Let us denote by
M the Fpsubmodule of Bunngenerated by P Ibn for those I = (ffl1; s1; : :;:ffl*
*k; sk)
with sk < 0; let us denote by M+ the Fpsubmodule generated by P Ibn for those
I = (ffl1; s1; : :;:fflk; sk) with sk 0 or k = 0; clearly Bunnis the internal *
*direct sum
M M+ . The map (1  P 0) sends P Ibn to P I(1  P 0)bn; it clearly sends M+
into M+ , and it follows from Propositions 11.3 and 12.2 that it sends M to M*
* .
Thus, it suffices to check that the composite is the identity on these submodul*
*es.
On M , f sends ffbn to ff(1 + P 0+ (P 0)2+ . .).bn. It follows that the comp*
*osite
sends ffbn to ff(1  P 0)(1 + P 0+ (P 0)2+ . .).bn = ffbn, and so the composite*
* is the
identity on M .
To see that the composite is the identity on M+ , it suffices to check it on *
*a stan
dard basis element, P Jbn, where J = (ffl1; s1; : :;:fflk; sk) is an admissible*
* sequence
with e(J) n and sk 0. Write I for the concatenation (J; (0; 0)). Observe that
I is admissible and e(I) = e(J) n, so
f(P J(1  P 0)bn) = f(P Jbn)  f(P Ibn) = f(P Jbn)  (P Jbn + f(P Jbn)) = P Jb*
*n:
It follows that the composite is the identity. _*
*__
Proof of Theorem 6.2.Let V denote the composite of the enveloping algebra func
tor and the functor () Fp Fp. Since this is the free functor from restricted
Fpmodules to graded commutative Fpalgebras, it preserves colimits. To avoid
confusion, let us note the (isomorphic) image of Bunnin Bunnunder the map (1P *
*0)
28 MICHAEL A. MANDELL
discussed above by In; by Proposition 12.5, Bunnis isomorphic as a restricted F*
*p
module to the direct sum In Aunn, and it follows that the ring VBun is isomorp*
*hic
to the ring VIn VAunn. We therefore obtain isomorphisms
Tor*;*VIn(Fp ; VBunn) ~=Fp VInV Bunn~=VAunn
where the first map is the projection from the torsion product to the tensor pr*
*oduct
and the second map is induced by V applied to the quotient map Bunn! Aunn.
On the other hand, Proposition 12.4 identifies H*EFp [n] as VBunn. It is wel*
*l
known that H*Kn = H*K(Z=pZ ; n) can be identified with VAunn. We see from
Proposition 11.5 that the map a from EFp [n] to C*Kn in the construction of Bn
induces on homology groups the map VBunn! VAunnobtained by applying V to
the quotient map Bunn! Aunn. Likewise, the map pn :EFp [n] ! EFp [n] in the
construction of Bn induces the map VBunn! VBunnobtained by applying V to the
map (1  P 0): Bunn! Bunn; in other words, we can identify the map induced by
pn on homology as the inclusion VIn ! VBunn. Corollary 3.6 provides a spectral
sequence that calculates the homology groups of the pushout Bn and that has as
its E2 term Tor*;*VIn(Fp ; VBunn). From the discussion of the last paragraph, w*
*e see
that this spectral sequence degenerates at E2 with no extension problems and_th*
*at
the map from Bn to C*Kn is a weak equivalence. __
13.Cell Algebras over Flat Operads
The purpose of this section is to present the proof of Theorem 2.8. The argum*
*ent
is no more complicated to describe in full generality, and so we present it her*
*e this
way. This should cause no confusion since the discussion in this section relat*
*es
to the main lines of argument in this paper only through Theorem 2.8. In this
section, k denotes a fixed but arbitrary commutative ground ring, and we consid*
*er
the following class of operads of differential graded kmodules.
Definition 13.1.For a ring R, we say that a differential graded right Rmodule M
is right Rflat if the functor M R () preserves quasiisomorphisms in the cate*
*gory
of differential graded left Rmodules, or equivalently, if the natural map from*
* the
differential torsion product Tor*R(M; N) to H*(M R N) is an isomorphism for
every differential graded left Rmodule N. We say that an operad of differential
graded kmodules F is flat if for each n, F(n) is right k[n]flat.
We have definitions of fibrations, cofibrations, and relative cell inclusions*
* of F
algebras completely analogous to those given in Section 2 for Ealgebras. We de*
*note
by F the free Falgebra functor on differential graded kmodules. The following
theorem is the main result we prove in this section.
Theorem 13.2. Let F be a flat operad of differential graded kmodules, let A b*
*e a
cofibrant Falgebra, and let Z be a differential graded kmodule. Then the cano*
*nical
map A ! A q FCZ is a quasiisomorphism.
Let A ! B be a map of Falgebras, and let F B be the free differential graded
kmodule with zero differential that has one generator zb in dimension n for ea*
*ch
element b of B in dimension n  1. Let xb denote the unique element of the
differential graded kmodule CF B whose differential is zb. We then have a map *
*of
differential graded kmodules F B ! B that sends xbto b for all b. The induced *
*map
of Falgebras AqFCF B ! B is a surjection and the canonical map A ! AqFCF B
is a relative cell inclusion and clearly has the left lifting property with res*
*pect to
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 29
fibrations. Thus, we obtain the following corollary of the previous theorem, of*
* which
Theorem 2.8 is the special case k = Fp, F = E.
Corollary 13.3.Let F be a flat operad of differential graded kmodules. Any map
of Falgebras f :A ! B can be factored functorially as f = q O j, where j is a
relative cell inclusion that has the left lifting property with respect to the *
*fibrations,
and q is a fibration. If A is cofibrant then j is in addition a quasiisomorphi*
*sm.
We begin the proof of Theorem 13.2 by noticing that the underlying differenti*
*al
graded kmodule of a coproduct of the form A q FX decomposes as a direct sum
of pieces homogeneous in X. We make this precise as follows.
Notation 13.4. For an Falgebra A, define UiA to be the differential graded k[i*
*]
module that makes the following diagram a coequalizer.
L _____//L
F(j + i) k[j](FA)(j)_____// F(j + i) k[j]A(j)_____//UiA
j0 j0
Here we understand the superscript (j) to denote the jth tensor over k power w*
*ith
(FA)(0)= A(0)= k. One map is induced by the Falgebra structure map FA ! A,
and the other is induced by the operadic multiplication of F.
Note that U0A is canonically isomorphic to A, and U1A is by definition the
"universal enveloping algebra" of A [12, 1.6.4]. More generally, the collection*
* UA =
{UnA} assembles into an operad with the universal property that the set of UA 
algebra structures on a differential graded kmodule X is naturally in onetoo*
*ne
correspondence with the set of pairs (; j) where :FX ! X is an Falgebra
structure on X and j :A ! X is a map Falgebras for this structure, c.f. [11,
1.18]. Our use for this construction is given by the following proposition.
Proposition 13.5.For a differential graded kmodule X, there is a natural iso
morphism of differential graded kmodules
L (i)
A q FX ~= UiA k[i]X :
Proof.A check of universal properties reveals that the coproduct on the left is*
* the
coequalizer of a pair of maps from F((FA) X) to F(A X). The proposition __
follows by a comparison of coequalizers. __
The inclusion of A into A q FX corresponds to the inclusion of the summand
U0A on the right hand side. Since (CZ)(i)is acyclic for all i > 0, Theorem 13.2*
* is
an immediate consequence of the following lemma.
Lemma 13.6. For a flat operad F and a cofibrant Falgebra A, UiA is right k[i]
flat for each i.
The remainder of this section is devoted to proving Lemma 13.6. We fix the
flat operad F and the cofibrant Falgebra A. We can assume without loss of
generality that A is a cell Falgebra, and so we can write A = ColimAn for some
cell Falgebras F(0) = A0 ae A1 ae . .,.degreewise free differential graded k
modules M1; M2; : :w:ith zero differential, and maps Mn+1 ! AnLsuch that An+1 =
An qFMn+1 FCMn+1. Let Nn = M1 . . .Mn, and let N = ColimNn = Mn.
For convenience, we understand N0 = M0 = 0.
Our argument for Lemma 13.6 is an inductive analysis of the following filtrat*
*ion
on the differential graded modules UiAn that generalizes the filtration given b*
*y the
direct sum decomposition of Proposition 13.5.
30 MICHAEL A. MANDELL
Notation 13.7. Let B be an Falgebra, let X be a differential graded kmodule,
and let g :X ! B be a map of differential graded kmodules. Let Umig be the
differential graded k[i]submodule of Ui(B qFX FCX) of elements of degree n or
less in CX, i.e. the submodule generated by the image of the elements
f xa1 . . .xaj2 F(j + i) (B CX)(j) for j 0
in which at most m of xa1; : :;:xaj 2 B CX map to a nonzero element under
the canonical projection B CX ! CX. We write UmiAn for Umig when g is the
given map Mn ! An1; we understand UmiA0 = UiA0 = F(i).
In order to understand this filtration, it is convenient to do some work in t*
*he
category of graded kmodules. Forgetting the differential, we can regard F as an
operad in the category of graded kmodules. We denote by F[the free functor from
graded kmodules to Falgebras of graded kmodules. To avoid confusion, we refer
to Falgebras of graded kmodules as F[algebras, reserving the term Falgebra
for Falgebras of differential graded kmodules. Note that when X is a differen*
*tial
graded kmodule, then the underlying F[algebra of FX is canonically isomorphic
to F[X.
Recall that for a differential graded kmodule X, the underlying graded kmod*
*ule
of CX is the direct sum of X and a copy of X shifted one degree down. We denote*
* by
oeX the graded ksubmodule of CX consisting of the shifted copy of X. Then since
An+1 = An qFMn+1 FCMn+1, we have that as F[algebras, An+1 = An q F[oeMn+1.
Passing to colimits, we obtain the following proposition.
Proposition 13.8.The map of graded kmodules oeN ! A induces an isomor
phism of F[algebras F[oeN ! A.
The differential that makes F[oeN into A is the obvious one determined by the
Leibniz rule and the operadic multiplication of F, writing the differential of *
*an
element of oeMn (the image of the corresponding element of Mn in An1) as an
element of F[oeN. We can give a description of the underlying graded k[i]module
of UiA, generalizing the description of A above. For i 0, let U[iA be the grad*
*ed
right k[i]module
L (j)
U[iA = F(j + i) k[j](oeN)
j0
A comparison of (now split) coequalizers gives the following result.
Proposition 13.9.The underlying graded k[i]module of UiA is canonically iso
morphic to U[iA.
An entirely analogous description of the underlying graded k[i]modules of
UiAn holds. More generally, we can describe the underlying graded k[i]module
of UmiAn as the graded submodule of UiAn generated by elements of
f xa1 . . .xaj2 F(j + i) k[j](oeNn)(j)
in which at most m of xa1; : :;:xaj map to a nonzero element under the canonic*
*al
projection oeNn ! oeMn. Since Mn is a direct summand of Nn, for n > 0 we can
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 31
identify the inclusion of UmiAn in UiAn as the map of graded k[i]modules
L1 max(j;m)L
F(j + i) k[jlxl] (oeNn1)(jl) (oeMn)(l)
j=0 l=0
1L Lj
! F(j + i) k[jlxl] (oeNn1)(jl) (oeMn)(l)
j=0l=0
~= L F(j + i) k[j](oeNn)(j):
j=0
This identification is vital to our argument; we use it through the following i*
*mme
diate consequence.
Proposition 13.10.For all i 0, m; n > 0, the inclusion Um1iAn ! UmiAn is a
split monomorphism of the underlying graded k[i]modules. We have an isomor
phism of differential graded k[i]modules
UmiAn=Um1iAn ~=Ui+mAn1 k[m ](CMn=Mn)(m):
Monomorphisms of differential graded modules that are split on the underlying
graded modules play an important role in the proof of Lemma 13.6, and so we
introduce the following terminology.
Definition 13.11.Let R be a ring and let f :L ! M be a map of differential
graded right Rmodules. We say that f is an almost split monomorphism if the
map of underlying graded right Rmodules is a split monomorphism.
If L ! M is an almost split monomorphism of differential graded right R
modules, then for any left Rmodule P , the sequence
0 ! L R P ! M R P ! (M=L) R P ! 0
is exact, and so induces a long exact sequence on homology groups. From this
observation and the definition of right Rflat, we obtain the following proposi*
*tion.
Proposition 13.12.Let L ! M be an almost split monomorphism of differential
graded right Rmodules. If any two of L, M, M=L is right Rflat then so is the
third.
We need one more observation on flat differential graded modules. The followi*
*ng
proposition is an easy consequence of the definition. We apply it below with R =
k[m+i], S = k[m ], and T = k[i] for i; m 0.
Proposition 13.13.Let R, S, and T be kalgebras, let S k T ! R be a map of
kalgebras, let L be a differential graded left Smodule, and let M be a differ*
*ential
graded right Rmodule. If M is right Rflat, R is right S k T flat, and L is r*
*ight
kflat, then M S L is right T flat.
Finally, we complete our argument with the proof of Lemma 13.6.
Proof of Lemma 13.6.By passage to the sequential colimit, it suffices to prove *
*that
UmiAn is right k[i]flat for each i; m; n 0. In the case n = 0, this is equiva*
*lent
to the assumption that F is a flat operad. Assume by induction that this holds *
*for
UmiAn1 for all i; m.
32 MICHAEL A. MANDELL
Since U0iAn = UiAn1 ~=ColimUmiAn1, it is right k[i]flat by the inductive
hypothesis. In general, for m > 0, the inclusion of Um1iAn in UmiAn is an almo*
*st
split monomorphism of differential graded right k[i]modules, and the quotient
UmiAn=Um1iAn ~=Um+iAn1 k[m ](CMn=Mn)(m)
is right k[i]flat by Proposition 13.13. Then by Proposition 13.12 and inductio*
*n_
on m, we conclude that UmiAn is right k[i]flat. __
14. Proof of Theorems 3.4 and 3.5
The proofs of Theorems 3.4 and 3.5 rely heavily on the work of the last sec
tion and we follow the conventions and notations introduced there. In particular
k is a commutative ground ring and F is a flat operad of differential graded k
modules, and we prove the theorems in this context. Of course, we do not expect
the homology of a coproduct to be the differential torsion product for an arbit*
*rary
flat operad, e.g. the operad for associative kalgebras, so for the generalizat*
*ion of
Theorem 3.4, we must restrict to operads that are also "acyclic": We say that an
operad F is acyclic if it comes equipped with an acyclic augmentation, a map of
operads F ! N that is a componentwise quasiisomorphism, where N is the operad
for commutative kalgebras, N (n) = k. We prove the following generalizations of
Theorems 3.4 and 3.5.
Theorem 14.1. Let F be an acyclic flat operad, and let B and C be cofibrant
Falgebras. The canonical map
F(2) B C ! F(2) (B C) (B C) ! B q C
induces an isomorphism Tor*k(B; C) ! H*(B q C).
Theorem 14.2. Let F be a flat operad, let A, B be cofibrant Falgebras, A ! B
a map of Falgebras and A ! C a cofibration of Falgebras. Then the canonical
map N(fi(B; A; C)) ! B qA C is a quasiisomorphism.
The proofs of these theorems consist of very similar arguments that study the*
* fil
trations described in 13.7 on the underlying differential graded module of a pu*
*shout
of Falgebras. We use the following observation many times in these arguments; *
*it
is an immediate consequence of the Tor version of the definition of a right Rf*
*lat
differential graded module.
Proposition 14.3.If X ! Y is a quasiisomorphism of right Rflat differential
graded modules and Z is any differential graded left Rmodule, then the map X R
Z ! Y R Z is a quasiisomorphism.
We begin with the proof of Theorem 14.1. The following proposition gives the
base case for the main part of the argument below.
Proposition 14.4.Let F be an acyclic flat operad, and let B be a cofibrant F
algebra. The natural map
F(2) (B F(i)) ! F(i + 1) B ! UiB
is a quasiisomorphism.
Proof.The natural map we have in mind is the composite of the map induced by
the operadic multiplication and the canonical map in Definition 13.4. The first*
* map
is a quasiisomorphism since F is acyclic and B is kflat by Lemma 13.6. Thus, *
*to
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 33
see that the composite is a quasiisomorphism, we only need to check that the m*
*ap
F(i + 1) B ! UiB is a quasiisomorphism.
We assume without loss of generality that B is a cell Falgebra; write B =
ColimBn where B0 = F(0) and Bn = Bn1 qFMn FCMn for some Mn as in
Definition 2.4. Since F(i + 1) B ~=ColimF(i + 1) Bn and UiB ~=ColimUiBn,
it suffices to show that the map F(i + 1) Bn ! UiBn is a quasiisomorphism for
each n. The case n = 0 follows from the assumption that F is flat and acyclic.
Assume by induction that this map is a quasiisomorphism for Bn1 for all i.
Write UmiBn for Umig as in 13.7 for g the given map Mn ! Bn1 above. The
map F(i + 1) Bn ! UiBn restricts to a map
F(i + 1) Um0Bn ! UmiBn:
In other words, F(i + 1) Bn ! UiBn is a filtered map. Consider the strongly
convergent spectral sequences associated to this filtered map. The induced map *
*on
E1terms consists of the maps
F(i + 1) (UsBn1 k[s](CMn=Mn)(s)) ! Us+iBn1 k[s](CMn=Mn)(s):
These map are quasiisomorphisms by Proposition 14.3 since each map F(i + 1)
UsBn1 ! Us+iBn1 is a quasiisomorphism by the inductive hypothesis. It follows
that the map of spectral sequences is an isomorphism from E2 onwards. The map
F(i + 1) Bn ~=Colimm F(i + 1) Um0Bn ! Colimm UmiBn ~=UiBn
is therefore a quasiisomorphism. ___
Proof of Theorem 14.1.We can assume without loss of generality that B and C are
cell Falgebras. Write C = ColimCn where C0 = F(0) and Cn = Cn1qFMn FCMn
for some Mn as in Definition 2.4. It suffices to prove that the map F(2)(BCn) !
B q Cn is a quasiisomorphism for each n. In fact, it is convenient for our ind*
*uctive
argument to prove that the map
F(2) (B UiCn) ! Ui(B q Cn)
is a quasiisomorphism for all i; n 0. In the case n = 0, this follows from t*
*he
previous proposition; assume by induction that this holds for Cn1 for all i.
Let UmiCn denote Umig as in 13.7 for g the given map Mn ! Cn1, and let
Umi(B q Cn) denote Umih for h the composite Mn ! Cn1 ! B q Cn1; we
understand UmiC0 = UiC0 = F(i) and Umi(B q C0) = Ui(B q C0) = UiB. The
map displayed above restricts to a map
F(2) (B UmiCn) ! Umi(B q Cn)
and induces a map of the strongly convergent spectral sequences associated to t*
*hese
filtrations. The map on E1terms consists of the maps
F(2) B Us+iCn1 k[s](CMn=Mn)(s)!
Us+i(B q Cn1) k[s](CMn=Mn)(s);
which are quasiisomorphisms by the inductive hypothesis and Proposition 14.3. *
*It
follows that the map of spectral sequences is an isomorphism from E2 onwards.
The map
F(2) (B UiCn) ~=Colimm F(2) (B UmiCn) !
Colimm Umi(B q Cn) ~=Ui(B q Cn)
34 MICHAEL A. MANDELL
is therefore a quasiisomorphism. ___
We now proceed with the proof of Theorem 14.2. As in the proof of Theorem 14.1
above, it is convenient to prove the more general result that (with the hypothe*
*sis
of Theorem 14.2) the natural map
N(Uifio(B; A; C)) ! Ui(B qA C)
is a quasiisomorphism for all i. We need the following observation for our arg*
*ument.
Proposition 14.5.Let F be a flat operad, let A ! B and A ! C be maps of
cofibrant Falgebras. Then N(Uifio(B; A; C)) is a right k[i]flat differential *
*graded
module.
Proof.For a differential graded left k[i]module X,
N(Uifio(B; A; C)) k[i]X ~=N(Uifio(B; A; C) k[i]X):
The proposition now follows from Lemma 13.6. ___
The proof of Theorem 14.2 begins with the following special case.
Proposition 14.6.Let F be a flat operad, and let A ! B be a map of cofibrant
Falgebras. The natural map N(Uifio(B; A; A)) ! UiB is a quasiisomorphism.
Proof.By the usual argument, the map of simplicial Falgebras fio(B; A; A) !
B is a homotopy equivalence. Applying the functor Ui, we have that the map
Uifio(B; A; A) ! UiB is a homotopy equivalence of simplicial differential graded
k[i]modules and so its normalization is a chain homotopy equivalence of_differ*
*_
ential graded k[i]modules. __
Proof of Theorem 14.2.We assume without loss of generality that B is a cell F
algebra and the map A ! C is a relative cell inclusion. Write C = ColimCn where
C0 = A and Cn = Cn1 qFMn FCMn for some differential graded kmodules Mn as
in Definition 2.4. It suffices to prove that the natural map
N(Uifio(B; A; Cn)) ! Ui(B qA Cn)
is a quasiisomorphism for all i; n 0. The case for C0 follows from the previo*
*us
proposition; assume by induction that this holds for Cn1.
Define Umi(BqA Cn) = Umig for g the composite map Mn ! Cn1 ! BqA Cn1.
Define Umifij(B; A; C) analogously. The simplicial map fio(B; A; Cn) ! B qA Cn
restricts to a simplicial map
Umifio(B; A; Cn) ! Umi(B qA Cn):
We take the normalization and consider the induced map on the strongly converge*
*nt
spectral sequences associated to these Umi filtrations. We can identify the map*
* on
E1terms as the map
N(Us+ifio(B; A; Cn1)) k[s](CMn=Mn)(s)!
Us+i(B qA Cn1) k[s](CMn=Mn)(s);
which is a quasiisomorphism by the inductive hypothesis and Proposition 14.3. *
*It
follows that the map of spectral sequences is an isomorphism from E2 onwards.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 35
The map
N(Uifio(B; A; Cn)) ~=Colimm N(Umifio(B; A; Cn)) !
Colimm Umi(B qA Cn) ~=Ui(B qA Cn)
is therefore a quasiisomorphism. ___
Appendix A. Other Fields
We use the techniques developed in the body of the paper to discuss when the
analogue of the Main Theorem holds for a field k. We prove the following theore*
*m.
In this theorem, denotes the Frobenius endomorphism on a field of positive
characteristic.
Theorem A.1. Let k be a field. The singular cochain functor with coefficients
in k induces an equivalence between the homotopy category of H*(; k)local [1]
nilpotent spaces of finite ktype and a full subcategory of the homotopy catego*
*ry of
E1 kalgebras if and only if k satisfies one of the following two conditions
(i)k = Q, the field of rational numbers.
(ii)k has positive characteristic and 1  is surjective.
It follows in particular that the analogue of the Main Theorem does not hold
when k is a finite field. The smallest field of characteristic p for which 1 *
* is
surjective is the fixed field in Fp of Z^pC Gal(Fp =Fp ).
For the finite fields Fq, we can be more specific about the the relationship *
*be
tween the padic homotopy category and the homotopy category of E1 Fq algebras.
The following theorem was suggested by W. G. Dwyer and M. J. Hopkins.
Theorem A.2. Let q = pn. For connected pcomplete nilpotent spaces of finite
ptype X and Y , there is a natural bijection
h EFq(C*(X; Fq); C*(Y ; Fq)) ~=H o(Y; X)
where hEFq denotes the homotopy category of E1 Fqalgebras and denotes the
free loop space functor.
Outline of the proof of Theorem A.1. For an arbitrary field k, there is no
difficulty in providing a natural Ekalgebra structure on the cochains of simpl*
*icial
sets, for some E1 koperad Ek. For example the work of [15] and the constructi*
*on
described in Section 1 produce such a structure. Write Ek for the category of E*
*k
algebras. We can form the adjoint functor U(; k) from Ekalgebras to simplicial
sets by the simplicial mapping set
Uo(A; k) = E (A; C*([ .]; k)):
Arguing as in Section 4, we obtain the following proposition.
Proposition A.3. The functors C*(; k) and U(; k) are contravariant right ad
joints between the category of Ekalgebras and the category of simplicial sets.*
* Their
right derived functors exist and give an adjunction between the homotopy catego*
*ry
of Ekalgebras and the homotopy category.
We say that a simplicial set is kresolvable if the unit of the derived adjun*
*c
tion X ! U(C*(X; k); k) is an isomorphism in the homotopy category. As an
elementary consequence of the previous proposition, we see that C*(; k) gives
an equivalence as in the statement of the theorem if and only if every connected
36 MICHAEL A. MANDELL
H*(; k)local nilpotent simplicial set of finite ktype is kresolvable. The b*
*ase field
Fp is irrelevant in Sections 25, and the arguments there apply to prove the fo*
*llow
ing propositions that allow us to argue inductively up principally refined Post*
*nikov
towers.
Proposition A.4. Let X = LimXn be the limit of a tower of Kan fibrations. As
sume that the canonical map from H*(X; k) to ColimH*(Xn; k) is an isomorphism.
If each Xn is kresolvable, then X is kresolvable.
Proposition A.5. Let X, Y , and Z be connected simplicial sets of finite ktype,
and assume that Z is simply connected. Let X ! Z be a map of simplicial sets,
and let Y ! Z be a Kan fibration. If X, Y , and Z are kresolvable, then so is *
*the
fiber product X xZ Y .
A connected space is nilpotent, H*(; k)local, and of finite ktype if and o*
*nly
if its Postnikov tower has a principal refinement with fibers:
(i)K(Q; n) when k is characteristic zero.
(ii)K(Z=pZ ; n) or K(Z^p; n) when k is characteristic p > 0.
By the argument in Section 6, K(Z^p; n) is easily seen to be kresolvable when
K(Z=pZ ; n) is. The theorem is therefore a consequence of the following two pro*
*po
sitions.
Proposition A.6. Let k be a field of characteristic zero. K(Q; n) is kresolvab*
*le
if and only if k = Q.
Proof.Write E for the free Ekalgebra functor. Let a: Ek[n] ! C*(K(Q; n); k) be
any map of Ekalgebras that sends the fundamental class of k[n] to the fundamen*
*tal
class of H*(K(Q; n); Q) H*(K(Q; n); k). Since k is characteristic zero, it is
easy to see that a is a quasiisomorphism, so the unit of the derived adjunction
is represented by the map K(Q; n) ! UEk[n]. It is straightforward to check that
UEk[n] is a K(k; n) and the map K(Q; n) ! K(k; n) induces on ssn the_inclusion_
Q k. __
Proposition A.7. Let k be a field of characteristic p > 0. K(Z=pZ ; n) is kre
solvable if and only if 1  is surjective on k.
Proof.We can construct a model Bn;kfor C*(Kn; k) exactly as in Section 6 and
prove that the map ffk: Bn;k! C*(Kn; k) is a quasiisomorphism just as in Sec
tion 12. We are therefore reduced to checking when the map Kn ! UBn;kis a
weak equivalence. Again, we have UBn;kgiven by a Kan fibration square
UBn;k _____//_UECk[n]
 
 
fflfflfflfflfflfflfflffl
UEk[n]__Upn_//UEk[n]:
The argument of Proposition 6.5 then applies to show that UEk[n] is a K(k; n) a*
*nd
the map Upn induces on ssn the map 1  . It follows that UBn;kis a K(Z=pZ ; n)
if and only if 1  is surjective. When 1  is surjective, it is straightforwa*
*rd_to
verify that the map Kn ! UBn;kis a weak equivalence. __
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 37
Outline of the proof of Theorem A.2. Let q = pn and consider the finite field
Fq. From the work above, it suffices to show that there is a natural isomorphism
X ! U(C*(X; Fq); Fq) in the homotopy category for X connected, pcomplete,
nilpotent, and of finite type ptype.
To make the argument, we need to assume that we have a map of operads of
Fpalgebras EFq Fq Fp ! E; we have such a map in the case when EFq and E
are constructed from the EilenbergZilber operad of [15] as outlined above and *
*in
Section 1. By changing E if necessary, we can assume without loss of generality
that this map is an isomorphism. Then we have an extension of scalars functor
E :EFq ! E defined by E() = () Fq Fp. The functor E preserves cofibrations
and quasiisomorphisms and is left adjoint to the forgetful functor that regard*
*s an
Ealgebra as an EFqalgebra. In particular, we have the following proposition.
Proposition A.8. There is a canonical natural isomorphism of simplicial sets
U(E()) ~=EFq(; C*[ .]).
Let = n denote the nth iterate of the Frobenius automorphism on Fp. Since
is a map of Fqalgebras, we obtain a map of simplicial EFqalgebras
C*[ .] ~=C*([ .]; Fq) Fq Fpid!C*([ .]; Fq) Fq Fp~=C*[ .]:
We obtain a natural automorphism on U(E). Thus, we can regard U(E) as a
functor from the category of EFqalgebras to the category of Zequivariant simp*
*licial
sets. We can regard U(; Fq) as a functor to the category of Zequivariant simp*
*licial
sets by giving U(; Fq) the trivial Zaction. The natural map U(; Fq) ! U(E)
induced by the inclusion C*([ .]; Fq) ! C*[ .] is then Zequivariant.
For a Zequivariant simplicial set X, let Xh be the homotopy equalizer of the
idand (where as above generates the Zaction): Let Xh be the simplicial set
that makes the following diagram a pullback.
Xh _____//_X[1]
 
 
fflffl fflffl
X __idx_//X x X
Since the natural transformation U(; Fq) ! U(E) factors through the fixed
points of , we obtain a natural map U(; Fq) ! U(E)h . We prove below the
following theorem.
Theorem A.9. The natural map U(A; Fq) ! U(EA)h is a weak equivalence
when A is cofibrant.
Theorem A.9 is the main fact needed for Theorem A.2.
Proof of Theorem A.2.Let X be a simplicial set, let A ! C*(X; Fq) be a cofibrant
approximation in the category of EFqalgebras, and let B ! C*X be a cofibrant
approximation in the category of Ealgebras. Since EA is cofibrant, we can choo*
*se
a map of Ealgebras A ! B so that the composite EA ! C*X coincides with the
composite of EA ! EC*(X; Fq) and the natural map of Ealgebras EC*(X; Fq) !
C*X. Then we have a composite map
X ! UB ! UEA;
38 MICHAEL A. MANDELL
natural in X in the homotopy category, which is a weak equivalence when X is co*
*n
nected pcomplete nilpotent of finite ptype by the Main Theorem. It is straigh*
*t
forward to check that the map X ! UEA factors through U(A; Fq), and so is
Zequivariant when we give X the trivial action. Consider the maps
U(A; Fq) ! (UEA)h Xh
By Theorem A.9, the first map is a weak equivalence. When X is a connected
pcomplete nilpotent Kan complex of finite ptype, the second map is a weak_equ*
*iv_
alence and Xh is a model for the free loop space X. __
For the proof of Theorem A.9, we recall the definition of a cosimplicial reso*
*lution
from [7]. For an object A of EFq, a cosimplicial resolution of A is a cosimplic*
*ial EFq
algebra Ao together with a quasiisomorphism A0 ! A such that A0 is cofibrant,
each coface map in Ao is an acyclic cofibration, and each map (d*; An) ! An+1 is
a cofibration, where (d*; An) is the object described in [7, 4.3]: the colimit *
*of the
diagram in EFq with objects
o For each i, 0 i n + 1, a copy of An labelled (di; An)
o For each (i; j), 0 i < j n + 1, a copy of An1 labelled (djdi; An1) (we
understand A1 = Fq).
and maps
o For each (i; j), 0 i < j n + 1, a map (djdi; An1) ! (dj; An) given by
the map di:An1 ! An.
o For each (i; j), 0 i < j n + 1, a map (djdi; An1) ! (di; An) given by t*
*he
map dj1:An1 ! An.
Although EFq is not a model category, the following analogues of the results of*
* [7,
x6] still hold.
Proposition A.10. Let Ao be a cosimplicial resolution. The functor EFq(Ao; )
from EFqalgebras to simplicial sets preserves fibrations and weak equivalences.
Proof.That EFq(Ao; ) preserves fibrations and acyclic fibrations follows from *
*the
standard arguments (omitted) in [7, x6]. Since EFq(Ao; ) preserves acyclic fi
brations, to see that it preserves all weak equivalences, it suffices to show t*
*hat it
preserves weak equivalences between cell EFqalgebras. Since for cell EFqalgeb*
*ras,
we can factor a map as an acyclic cofibration followed by a fibration, we can_a*
*pply_
the dual of the argument for K. Brown's lemma [9, 9.9]. __
Proposition A.11. Let k = Fq or Fp. For any cosimplicial resolution of EFq
algebras Ao, the maps of simplicial sets
EFq(Ao; k) ! diagEFq(Ao; C*([ .]; k)) EFq(A0; C*([ .]; k))
are weak equivalences.
Proof.Since all the face maps of C*([ .]; k) are acyclic fibrations, the first *
*map is
a weak equivalence by the previous lemma. The simplicial Ekalgebra C*([ .]; k)
has the dual property that mapping into it converts acyclic cofibrations to acy*
*clic_
Kan fibrations, and so the second map is a weak equivalence. _*
*_
Proof of Theorem A.9.Since the weak equivalences in Proposition A.11 are Z
equivariant maps of Kan complexes (where for k = Fq we understand the trivial
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 39
action), it suffices to show that the map
EFq(Ao; Fq) ! EFq(Ao; Fp)h
is a weak equivalence. Factor the diagonal map Fp ! Fp x Fp as an acyclic
cofibration Fp ! P composed with a fibration P ! Fp x Fp, and let Q be the
EFqalgebra that makes the following diagram a pullback.
Q _____________//_B
 
 
fflfflfflffl fflfflfflffl
Fp __________//_Fx F
idx p p
The unit map Fq ! Q is a weak equivalence, and so the map EFq(Ao; Fq) !
EFq(Ao; Q) is a weak equivalence. Since EFq(Ao; ) preserves pullbacks and fibr*
*a
tions, we have that the following diagram is the pullback of a Kan fibration.
EFq(Ao; Q)___________//_EFq(Ao; P )
 
 
fflfflfflffl fflfflfflffl
EFq(Ao; Fp)idx__//EFq(Ao; Fp) x EFq(Ao; Fp)
Choosing a diagonal lift in the following diagram
EFq(Ao;fFp)______~______//flfflEFq(Ao;5P5)_
_______________
~ ______________ 
fflffl_________ fflfflfflffl
EFq(Ao; Fp)[1]____//_EFq(Ao; Fp) x EFq(Ao; Fp)
we obtain a weak equivalence EFq(Ao; Fp)h ! EFq(Ao; Q) factoring the weak
equivalence EFq(Ao; Fq) ! EFq(Ao; Q) above through the map EFq(Ao; Fq) !
EFq(Ao; Fp)h . ___
Appendix B. ProCategories and pProFinite Completion
In this section we describe the relation between the unit of the derived adju*
*nction
X ! UC*X and pprofinite completion in the sense of Sullivan [28, x3], [24, x2*
*.1].
The idea that there should be some relation was first suggested by W. G. Dwyer.
We prove the following theorem.
Theorem B.1. For any connected simplicial set X, the composite map
X ! UC*X ! U(C*(X; Fp) Fp Fp)
is pprofinite completion.
Here we are giving C*(X; Fp)FpFp the structure of an Ealgebra via the natural
isomorphism C*(X; Fp)FpFp ~=C*cont^X= ColimC*Xff, where ^X= {Xff} denotes
the "completion of X" [24, x2.1], [14, 1.2.2], the projective system of levelwi*
*se
finite quotients of X. The system of maps X ! Xffinduces a map of Ealgebras
C*(X; Fp) Fp Fp ! C*X that induces the map UC*X ! U(C*(X; Fp) Fp Fp)
above.
In other words, for the theorem above, we have used a version of the cochain
functor that factors through the category of profinite simplicial sets. From *
*this
40 MICHAEL A. MANDELL
perspective, it is clear that the theorem we should try to prove is that the fu*
*nctor
C*contfrom the category of profinite simplicial sets to the category of Ealge*
*bras is
a Quillen adjoint (to "U^") and that the natural transformation X ! ^ULC*contX *
*is
pprofinite completion in the sense of Morel [24, x2.1], where L is some cofib*
*rant
approximation functor. Unfortunately, this is not true; there is no adjoint fun*
*ctor
^Ufrom the category of Ealgebras to the category of profinite simplicial sets*
*. To
see this, note that the set of maps of profinite simplicial sets from the (con*
*stant)
standard simplex [n] to any profinite simplicial set is naturally a compact sp*
*ace,
and so the set of maps from an Ealgebras A to C*cont[n] ~=C*[n] would have
to be a compact space with an action of E (A; A) through continuous maps. On
the other hand, E (EFp [n]; C*[n]) ~=Fp is countable and Fp E (EFp [n]; EFp [n*
*])
acts transitively.
If we look at a larger category, the procategory of simplicial sets, then an*
* adjoint
functor does exist. Letting proS denote the procategory of simplicial sets, t*
*he
natural cochain functor C*cont:proS ! E to consider is the functor that takes a
pro simplicial set X = {Xff} to the Ealgebra ColimC*Xff. We prove the following
lemma below.
Lemma B.2. The functor C*cont:proS ! E has a right adjoint Uc, i.e. there is
a bijection proS (X; UcA) ~=E (A; C*contX), natural in pro simplicial sets X a*
*nd
Ealgebras A.
For the proof of Lemma B.2, we consider the functor C* from proS to indE ,
the indcategory of Ealgebras, the opposite category of the procategory of E *
*op.
The functor U :indE ! proS is a right adjoint to C*. We have an obvious funct*
*or
Colim:indE ! E , and C*cont= ColimC*. Of course Colimis a left adjoint (to the
constant functor), but in fact it is also a right adjoint. Lemma B.2 is an imme*
*diate
consequence of the following proposition, setting UcA = UcA.
Proposition B.3. The functor Colim :indE ! E has a left adjoint c: E !
indE .
The proof of the previous proposition is easy, but requires the following ter*
*mi
nology.
Definition B.4.We say that an Ealgebra A is compact if for any B = {Bff} in
indE , the natural map ColimE (A; Bff) ! E (A; ColimB) is a bijection. We say
that an Ealgebra A is finitely presented if A a coequalizer (in E )
EN _____////_EM__//_A
for finitely generated differential graded Fpmodules M and N.
Clearly EM is compact when M is finitely generated, and so finitely presented
Ealgebras are compact. For an arbitrary Ealgebra consider the category RA who*
*se
objects consist of ordered pairs (M; N) where M is a finitely generated differe*
*ntial
graded submodule of A and N is a finitely generated differential graded submodu*
*le
of EM sent to zero under the induced map EM ! A; the maps in RA are the
inclusions. We have a functor DA from RA to finitely presented Ealgebras, send*
*ing
(M; N) to the coequalizer (in E ) of the maps EN ! EM induced by the inclusion
N ! EM and the zero map N ! EM. The category RA is filtered, and c() =
D() specifies a welldefined functor from E to indE . Since the canonical map
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 41
ColimRA DA ! A is an isomorphism, we have that for any B = {Bff} in indE ,
E (A; ColimB) ~=E (Colim cA; ColimB) ~=LimRA E (DA ; ColimB)
~=LimRA Colim(DA ; Bff) = indE (cA; B)
This proves Proposition B.3. We find it useful to note here the following easy
observations.
Proposition B.5. A finite cell Ealgebra is finitely presented.
Proposition B.6. The functor c() is an equivalence between E and the full sub
category of indE consisting of the inductive systems of compact Ealgebras.
To take avantage of the adjoint functor Uc, we need a homotopy theory for the
catgeory proS of pro simplicial sets. This theory is provided in the recent wo*
*rk of
Isaksen [16], where it is shown that the category proS is a closed model categ*
*ory.
Following the terminology there, say that a map f :X ! Y is a level map if X and
Y are indexed on the opposite of the same filtered category I and f is represen*
*ted
by a map of diagrams on I op. A map f :X ! Y in proS is a strong weak
equivalence if it is a level map where for all n 0, fi 2 I , there exists ff !*
* fi in
I opsuch that for every choice of basepoint in Xff, there is a map ssnYfi! ssnX*
*ff
that makes the following diagram commute.
fff
ssnXff____//ssnYff
 vvvv 
 vvv 
fflfflvvfflffl
ssnXfi_f__//ssnYfi
fi
A weak equivalence in proS is a map in proS that is isomorphic to a strong we*
*ak
equivalence. It is proved in [16] that a level map is a weak equivalence if and*
* only
if it is a strong weak equivalence. Thus, since every map in proS is isomorphic
to a level map, when X is a pro connected based simplicial set, a map X ! Y is
a weak equivalence if and only if it induces a proisomorphism of each homotopy
progroup {ssnXff} ! {ssnYfi}.
The cofibrations are the maps isomorphic to level maps that are level cofibra
tions; in particular all objects are cofibrant. It is shown that the constant *
*pro
simplicial set on a Kan simplicial set with only finitely many nontrivial homot*
*opy
groups is fibrant in proS and a Kan fibration between such simplicial sets is a
fibration in proS . It follows that we can identify the functor C*contX as the*
* set
of maps from X to K(Fp ; n) in the homotopy category of proS . Thus, C*cont
converts cofibrations to fibrations and preserves weak equivalences. As an imed*
*iate
consequence of Theorems 2.14 and 2.15, we obtain the following proposition.
Proposition B.7. The (right) derived functor Uc of Uc exists and gives and ad
junction hE (A; C*contX) ~=proS (X; UcA).
The functor C* from the homotopy category to the homotopy category of E
algebras factors as the composite of the constant functor and C*cont, and so it
follows that the functor U is the composite of Uc and the right derived functor*
* of
Lim. The forgetful functor from Morel's model category of profinite simplicial*
* sets
to Isaksen's model category of pro simplicial sets is a right adjoint that pres*
*erves
fibrations and acyclic fibrations, and so the right derived functor of Lim from*
* the
42 MICHAEL A. MANDELL
homotopy category of profinite simplicial sets to the homotopy category is the
composite of the right derived functor of the forgetful functor and the right d*
*erived
functor of Lim from the homotopy category of pro simplicial sets to the homotopy
category. Since profinite completion in the sense of Sullivan is the composit*
*e of
the completion functor from simplicial sets to profinite simplicial sets and t*
*he right
derived functor of Lim [24, x2.1], Theorem B.1 is an immediate consequence of t*
*he
following lemma.
Lemma B.8. Let X be a connected simplicial set. There is a fibrant profinite
simplicial set Y , a weak equivalence of profinite simplicial sets ^X! Y , and*
* a cofi
brant approximation A ! C*contY such that the map Y ! UcA is a weak equivalence
of pro simplicial sets.
The remainder of the section is devoted to the proof of Lemma B.8. According *
*to
[24, x2.1], we can take Y = {Yff} to have the property that each Yffis a connec*
*ted
"pespace finis", i.e. has finitely many nontrivial homotopy groups, all of wh*
*ich are
finite pgroups. Choose such a Y and write I for the filtering category opposite
to the category that indexes Y . It is not hard to see that we can make an I
diagram of cofibrant Ealgebras Affwith a natural acyclic fibration Aff! C*Yff
and with the property that A = Colim Affis also cofibrant. For example, it is
straightforward to check that LC*Yffhas this property where L is the cofibrant
approximation functor obtained by the small object argument in Proposition 2.6.
Alternatively, after replacing Y with an isomorphic object if necessary, we can
assume that I is a cofinite strongly directed category, and then such a diagram
Affis easily constructed by induction. Note that however the Affare constructed,
the induced map A ! C*contY is an acyclic fibration. We choose Y and A in this
way in order to make the following observation.
Proposition B.9. For Y and A as above, for each ff, the map from the constant
pro simplicial set Yffto UcAffis a weak equivalence.
Proof.According to Remark 7.4, since Yffhas only finitely many nontrivial homo
topy groups, all of which are finite pgroups, there is a finite cell Ealgebra*
* B and a
quasiisomorphism B ! C*Y . By Proposition B.5, UcB is isomorphic to the con
stant pro simplicial set on UB, and so by the Main Theorem, the map Yff! UcB
is a weak equivalence. But by the left lifting property, the map B ! C*Y can be
factored through a quasiisomorphism B ! Aff, and so the map Y ! UcAffis_also_
a weak equivalence. __
Let J be the category whose set of objects is the disjoint union of the sets
of objects of the RAffwhere ff ranges over the objects of I . For a 2 RAff, b 2
RAfi, we have a map a ! b in J for each map Aff! Afiin I that maps the
pair of differential graded submodules (M; N) corresponding to a into the pair *
*of
differential graded submodules corresponding to B. Clearly J is a filtered cate*
*gory.
The functors DAff:RAff! E assemble to a functor D :J ! E , which we regard
as an element of indE . We have a canonical map D ! {Aff} covering the forgetf*
*ul
functor J ! I and inducing an isomorphism ColimD ! ColimAff= A. Since
D is a diagram of compact Ealgebras, the map D ! {Aff} factors through an
isomorphism D ! cA by Proposition B.6.
Proof of Lemma B.8.If we choose a basepoint for X, we obtain compatible base
points for the Yffso that Y is a system of based connected simplicial sets. Then
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 43
it suffices to show that the map Y ! UcA induces a proisomorphism of each ho
motopy progroup ssnY ! ssnUcA. By construction, the map Y ! UcA factors
through the map Y ! {UAff}; we base UAffand the simplicial sets in UcAffat
the image of the basepoint of Yff. Looking at D, we can identify ssnUcA as the
limit (over ff in I ) in progroups of the progroups {ssnUcAff}. Since ssnY is*
* the
limit (over ff in I ) in progroups of the constant progroups ssnYff, the_lemm*
*a_now
follows from Proposition B.9. __
Appendix C. E1 Ring Spectra under HFp
We sketch how the arguments in this paper can be modified to prove the follow*
*ing
unpublished theorem of W. G. Dwyer and M. J. Hopkins [8] comparing the padic
homotopy category with the homotopy category of E1 HFp ring spectra.
Theorem C.1. (DwyerHopkins) The free mapping spectrum functor F (()+ ; Fp)
induces an equivalence between the homotopy category of connected pcomplete ni*
*lpo
tent spaces of finite ptype and a full subcategory of the homotopy category of*
* E1
HFp ring spectra.
By the homotopy category of E1 HFp ring spectra, we mean the category
obtained from the category of E1 ring spectra under the (cofibrant) E1 ring
spectrum HFp by formally inverting the weak equivalences. The free mapping
spectrum F (X+ ; HFp ) is naturally an E1 ring spectrum with an E1 ring map
HFp = F (*+ ; HFp ) ! F (X+ ; HFp )
induced by the collapse map X ! *. The functor F (()+ ; Fp) therefore takes
values in the category of E1 HFp ring spectra. This functor is the spectrum an*
*a
logue of the singular chain complex. Its right derived functor represents unred*
*uced
ordinary cohomology with coefficients in Fp in the sense that there is a canoni*
*cal
map ss*(F (X+ ; Fp)) ! H*(X; Fp) that is an isomorphism if X is a CW complex.
It is convenient for us to use a modern variant of the category of E1 HFp ri*
*ng
spectra, the category of commutative HFp algebras, a certain subcategory intro
duced in [10]. The forgetful functor from commutative HFp algebras to E1 HFp
ring spectra induces an equivalence of homotopy categories. We have a commuta
tive HFp algebra variation of the free mapping spectrum functor, given by
F X = S ^L F (X+ ; HFp ):
There is a natural map F X ! F (X+ ; HFp ) that is always a weak equivalence,
and so it suffices to prove that the functor F induces an equivalence between t*
*he
homotopy category of connected pcomplete nilpotent spaces of finite ptype and
a full subcategory of the homotopy category of commutative HFp algebras. We
denote the category of commutative HFp 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 hC .
The commutative HFp algebra F X is the "cotensor" of HFp with X [10, VII.2.9*
*].
In general, the cotensor AX of a commutative HFp algebra A with the space
X is the commutative HFp algebra that solves the universal mapping problem
C (; AX ) ~=U (X; C (; A)), where U denotes the category of (compactly gen
erated and weakly Hausdorff) spaces. Similarly, the tensor A X of A with the
space X is the commutative HFp algebra that solves the universal mapping prob
lem C (A X; ) ~=U (X; C (A; )). Clearly, when they exist AX and A X are
44 MICHAEL A. MANDELL
unique up to canonical isomorphism, and [10, VII.2.9] guarantees that they exist
for any A and any X. The significance of the identification of F X as the tenso*
*r is
in the following proposition.
Proposition C.2. The functor T :C ! U defined by T A = C (A; HFp ) is a
continuous contravariant right adjoint to F . In other words, there is a homeo
morphism U (X; T A) ~=C (A; F X), natural in the space X and the commutative
HFp algebra A.
We have introduced the notion of tensor here to take advantage of [10, VII.4.*
*16]
that identifies the tensor A I as a Quillen cylinder object when A is cofibran*
*t.
This allows us to relate the homotopies in the sense of Quillen with topological
homotopies defined in terms of () I or in terms of paths in mapping spaces. In
particular, since all objects in C are fibrant, it follows that the natural tra*
*nsforma
tion ss0(C (A; )) ! hC (A; ) is an isomorphism when A is cofibrant. Since the
adjunction isomorphism U (X; T A) ~=C (A; F X) is a homeomorphism, letting X
vary over the spheres, we obtain the following proposition.
Proposition C.3. The functor T preserves weak equivalences between cofibrant
objects.
As a slight generalization of the proof of [10, VII.4.16], it is elementary t*
*o check
that when A is a cofibrant object of C and A ! B is a cofibration, the map
(A I) qA B ! B I is an acyclic cofibration and therefore (since every object
is fibrant) the inclusion of a retract. Since T also converts pushouts to pullb*
*acks,
applying T and using the tensor adjunction, we obtain the following proposition.
Proposition C.4. The functor T converts cofibrations to fibrations.
The functors F and T 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 o(X; TA) ~=hC (A; FX).
For the purposes of this section, let us say that a space X is HFp resolvabl*
*e if
the unit of the derived adjunction X ! TFX is a weak equivalence. Thus, we need
to show that if X is a connected pcomplete nilpotent space of finite ptype, t*
*hen X
is HFp resolvable. Again, we work by induction up principally refined 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 Proposition 4.4.
Proposition C.6. Let X = Lim Xn be the limit of a tower of Serre fibrations.
Assume that the canonical map from H*X to ColimH*Xn is an isomorphism. If
each Xn is HFp resolvable, then X is resolvable.
We have in addition the following analogue of Theorem 1.2.
Theorem C.7. Let X, Y , and Z be connected spaces of finite ptype, and assume
that Z is simply connected. Let X ! Z be a map, and let Y ! Z be a Serre
fibration. If X, Y , and Z are HFp resolvable, then so is the fiber product X *
*xZ Y .
The proof of this theorem is essentially the same in outline as the proof of *
*The
orem 1.2. The analogue of Lemma 5.2 can be proved by observing that the bar
construction of the cofibrant approximations in C is equivalent to the (thicken*
*ed) re
alization of F applied to the cobar construction of the singular simplicial set*
*s on the
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 45
spaces Xo, Yo, and Zo. Some fiddling with the filtration induced by the cosimpl*
*icial
direction of the cobar construction and the filtration induced by the skeletal *
*filtra
tion of the singular simplicial sets allows the identification of TorFZo*(F *
*Xo; F Yo)
as Tor*C*Z(C*X; C*Y ) and the composite map
Tor*C*Z(C*X; C*Y ) ~=TorFZo*(F Xo; F Yo) ! ss*F (Xo xZo Yo) ~=H*(XxZY*
* )
as the EilenbergMoore map.
To complete the proof of Theorem C.1, we need to see that K(Z=pZ ; n) is HFp 
resolvable. It then follows as in Section 1.3 that K(Z^p; n) is HFp resolvable*
* and by
induction up principal Postnikov towers that every connected pcomplete nilpote*
*nt
space of finite ptype is HFp resolvable. The remainder of the appendix is dev*
*oted
to sketching a proof of the following theorem.
Theorem C.8. For n 1, K(Z=pZ ; n) is HFp resolvable.
The homotopy groups of a commutative HFp algebra have an action by the
algebra B, and it is elementary to show that for the "free" commutative HFp 
algebra on the spectrum Sn , denoted PSnHFpin [10], ss*PSnHFpis V Bunn, the
extended Fpalgebra on the enveloping algebra of the free unstable Bmodule on
one generator in degree n. We construct a commutative HFp algebra Bn as the
commutative HFp algebra that makes the following diagram a pushout in C .
PSnFp____//PCSnFp
pn 
fflffl fflffl
PSnFp______//Bn
Here pn is any map in the unique homotopy class that on homotopy groups sends
the fundamental class of ssn SnHFpto 1  P 0applied to the fundamental class.
Choosing a map a: PSnFp! F K(Z=pZ ; n) that represents the fundamental class
of Hn(K(Z=pZ ; n)), and a null homotopy PCSnFp! F K(Z=pZ ; n) for the map
pn O a: PSnFp! F K(Z=pZ ; n), we obtain an induced map Bn ! F K(Z=pZ ; n).
Proposition C.9. For n 1, the map Bn ! F K(Z=pZ ; n) is a weak equivalence.
The proof uses the EilenbergMoore spectral sequence of [10, IV.4.1] in place*
* of
the EilenbergMoore spectral sequence of Section 3, but otherwise is the same as
the proof of Theorem 6.2.
Since Bn is a cofibrant commutative HFp algebra, the unit of the derived adj*
*unc
tion is represented by the map K(Z=pZ ; n) ! T Bn adjoint to the map constructed
above. Since Bn is defined as a pushout of a cofibration, Proposition C.4 allo*
*ws
us to identify T Bn as the pullback of fibration. Looking at the mapping spaces
and using the freeness adjunction, we see that T Bn is the homotopy fiber of an
endomorphism on K(Fp ; n). Write ffn for the induced endomorphism on Fp. To
see that T Bn is a K(Z=pZ ; n), it suffices to show that ffn is 1  . Once we k*
*now
that T Bn is a K(Z=pZ ; n), the argument of Corollary 6.3 shows that the map
K(Z=pZ ; n) ! T Bn is a weak equivalence, completing the proof of Theorem C.8.
Unfortunately, the simple argument given in Proposition 6.5 to identify ffn as
1  in the algebraic case does not have a topological analogue. Here we must
46 MICHAEL A. MANDELL
use homotopy theory to make this identification. The key observation is that the
commutative HFp algebras Bn are related by "suspension". We make this precise
in the following proposition. For this proposition, note that the definition of*
* Bn
makes sense for n = 0, although the map B0 ! F K(Z=pZ ; 0) may not be a weak
equivalence.
Proposition C.10. For n > 0, Bn1 is homotopy equivalent as a commutative
HFp algebra to the pushout of the following diagram
Bn _____//_Bn S1


fflffl
HFp
where the map Bn ! HFp is the augmentation Bn ! F K(Z=pZ ; n) ! F * = HFp
induced by the inclusion of the basepoint of K(Z=pZ ; n) and the map Bn ! B S1
is induced by the inclusion * ! S1.
For an augmented commutative HFp algebra A, denote the analogous pushout
for A as C A. If we give PSnHFpthe augmentation induced by applying P to the
map SnHFp! *, then C PSnHFpis canonically isomorphic to PSn+1HFp. This gives
us a canonical suspension homomorphism oe :"ssnA ! "ssn+1C A, where "ss*is
the kernel of the augmentation map ss*A ! ss*HFp . The following proposition is
closely related to and can be deduced from [22, 3.3].
Proposition C.11. The suspension homomorphism oe commutes with the opera
tion P sfor all s.
We can choose the map pn in the construction of Bn to be augmented for the
augmentation described on PSnHFpabove. Then it follows from the previous propo
sition that C pn is homotopic to pn1. This observation can be used to prove
Proposition C.10.
It follows from Proposition C.10 that T Bn1 is the loop space of T Bn. In fa*
*ct,
we see from the discussion above that the fiber sequence for T Bn1
T Bn1 ! K(Fp ; n  1) ! K(Fp ; n  1)
is the loop of the corresponding fiber sequence for T Bn. In particular, ffn a*
*nd
ffn1 are the same endomorphisms of Fp. Since P 0performs the pth power map
on classes in degree zero, ff0 is 1  . We conclude that ffn is 1  .
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Department of Mathematics, M. I. T., Cambridge, MA
Email address: mandell@math.mit.edu