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: January 26, 1998, 17:28
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 Quillen, Sullivan, 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 subcategory of t*
*he
homotopy category of commutative differential graded Qalgebras, and that the
functor underlying this equivalence is closely related to the singular cochain *
*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, it would be n*
*aive
to think that any reasonably useful subcategory of the padic homotopy category
could be equivalent to a category of commutative differential graded algebras. *
*We
must instead look to a more sophisticated class of algebras, E1 algebras [12].*
* In
fact, it turns out that even the category of E1 Fpalgebras is not quite suffi*
*cient;
rather we consider E1 algebras over the algebraic closure Fp of Fp. We prove t*
*he
following theorem.
Main Theorem. The singular cochain functor with coefficients in Fp induces a
contravariant equivalence from the homotopy category of connected nilpotent p
complete spaces of finite ptype to a full subcategory of the homotopy category*
* 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. We remind the reader that a conne*
*cted
space is pcomplete, nilpotent, and of finite ptype if and only if its Postnik*
*ov tower
has a principal refinement in which each fiber is of type K(Z=pZ ; n) or K(Z^p;*
* n),
where Z^pdenotes the padic integers.
____________
Date: January 26, 1998, 17:28.
1991 Mathematics Subject Classification. Primary 55P15; Secondary 55P60.
The author was supported by an Alfred P. Sloan Dissertation Fellowship.
1
2 MICHAEL A. MANDELL
By the homotopy category of E1 algebras, we mean the category obtained from
the category of algebras over a particular but unspecified E1 Fp operad by form*
*ally
inverting the maps in that category that are quasiisomorphisms of the underlyi*
*ng
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 [12, I] for a good introduction to op*
*erads,
E1 operads, and E1 algebras.
Comparison with Other Approaches. The papers [9, 11, 20] and the unpub
lished ideas of [6] all compare padic homotopy theory to various homotopy cate
gories of algebras (or coalgebras). We give a short comparison of these results*
* to
the results proved here.
The first announced results along the lines of our Main Theorem appeared in
[20]. The arguments there are not well justified, however, and some of the resu*
*lts
appear to be wrong.
More recently, [9, 11] have compared the padic homotopy category with the
homotopy categories of simplicial cocommutative coalgebras and cosimplicial com
mutative algebras. In particular, [9] proves that the padic homotopy category
embeds as a full subcategory of the homotopy category of cocommutative simpli
cial Fpcoalgebras. It is straightforward to describe the relationship between *
*this
theorem and our Main Theorem. There is a functor from the homotopy category
of simplicial cocommutative coalgebras to the homotopy category of E1 algebras
given by normalization of the dual cosimplicial commutative algebra [10] (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 [11, 6.3].
The unpublished ideas of [6] 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
Theorem. We sketch the argument in Appendix B. A direct comparison between
this approach and our Main Theorem 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 grad*
*ed
Fpalgebras as the singular cochain functor. We will provide this comparison and
this identification in [14] and [15].
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 the field k.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 3
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 [10] 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. Unfortunate*
*ly,
Z is not an E1 operad since it is not free and since it is nonzero in both po*
*sitive
and negative degrees. Nevertheless, when we take coefficients in a field, both *
*these
deficiencies are trivial to overcome. Let Z be the "(co)connective cover" of *
*Z:
Z(n) is the differential graded Fpmodule that is equal to Z(n) in degrees less
than zero, equal to the kernel of the differential in degree zero, and zero in *
*positive
degrees. The operadic multiplication of Z lifts to Z, making it an acyclic oper*
*ad.
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 structu*
*re
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
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 in Section 2 that this
functor has a right adjoint U. The functor U provides the inverse equivalence in
the Main Theorem.
Precisely, the adjoint U is a contravariant functor from the homotopy category
of E1 Fp algebras 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 adjunction X ! UC*X. For the purposes of this paper,
we say that a simplicial set X is resolvable by E1 Fpalgebras or just resolva*
*ble if
the map uX is an isomorphism in the homotopy category. In Section 3, 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 .
4 MICHAEL A. MANDELL
These theorems allow us to argue inductively up towers of principal Kan fibra
tions. The following theorem proved in Section 4 provides a base case.
Theorem 1.3. K(Z=pZ ; n) and K(Z^p; n) are resolvable for n 1.
We conclude that every connected nilpotent pcomplete 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 nilpotent pcomplete simplicial sets of finite ptype.
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 [17] 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 cochain complex *
*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.
2. Construction of the Functor U
In this section, we construct the functor U whose restriction provides the in*
*verse
equivalence of the Main Theorem. In fact, U is constructed as the derived funct*
*or
of a functor from the category of Ealgebras to the category of simplicial sets*
* that
we also denote as U. We begin by observing that the cochain functor C* from
simplicial sets to Ealgebras is an adjoint.
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 (Xn; C [m]):
n f :m!n
f in op
By construction M(X; C*) is an Ealgebra, contravariantly functorial in the sim
plicial set X.
Proposition 2.1.The cochain functor C* is canonically naturally isomorphic to
M(; C*) as a functor from simplicial sets to Ealgebras.
Proof.For each element of Xm , there is a canonical map [m] ! Xm , and the
collection of all such maps induces a map of Ealgebras
Q *
C*X ! P (Xm ; C [m]):
m
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 5
By naturality, this map factors through the equalizer to induce a map of Ealge*
*bras
C*X ! M(X; C*). The underlying differential graded kmodule of an equalizer of
Ealgebras is the equalizer of the underlying differential graded kmodules. It*
* follows
that the induced map C*X ! M(X; C*) is an isomorphism of the underlying *
* __
differential graded kmodules and therefore an isomorphism of Ealgebras. *
*__
The description of C* given by Proposition 2.1 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]))
~=E (A; Q C*[n]) = E (A; P (Xm ; C*[n])):
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 2.2.The functors U and C* are contravariant right adjoints between
the category of simplicial sets and the category of Ealgebras.
In [13, x5], we studied the homotopical properties of adjoint functors betwee*
*n a
closed model category and a category of E1 algebras. Since the discussion there
was in terms of covariant functors, we apply it to U, C* viewed as an adjoint p*
*air
between the category of Ealgebras and the opposite to the category of simplici*
*al
sets. As such, U is the left adjoint. Taking the closed model category structur*
*e on
the opposite category of simplicial as the one opposite to the standard one [19*
*] on
the category of simplicial sets, the "fibrations" are the maps opposite to mono*
*mor
phisms and the "weak equivalences" are the maps opposite to weak equivalences.
It follows that the functor C* converts "fibrations" to surjections and "weak e*
*quiv
alences" to quasiisomorphisms. It then follows from [13, 1.9,1.10] that the l*
*eft
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 contravariant functor, this
derived functor is the right derived functor, and we obtain the following propo*
*sition.
Proposition 2.3.The (right) derived functor of U exists and gives an adjunction
hE (A; C*X) ~=H o(X; UA).
Applying [13, 1.10] again, we obtain the following proposition, which is need*
*ed
in the proofs of Theorems 1.1 and 1.2 in the next section.
Proposition 2.4.The functor U converts relative cell inclusions of Ealgebras to
Kan fibrations of simplicial sets.
According to [13, 1.9], the derived functor of U is constructed by first appr*
*oxi
mating an arbitrary Ealgebra with a cell Ealgebra [13, 1.2] and then applying*
* U.
This gives us the following proposition.
6 MICHAEL A. MANDELL
Proposition 2.5.Let X be a simplicial set and A ! C*X a quasiisomorphism,
where A is a cell 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 2.6.The functor U converts relative cell inclusions to H*(; Fp)
local fibrations. For a cell Ealgebra A, UA is an H*(; Fp)local simplicial s*
*et.
3. 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 cell Ealgebra approximations and applying Proposition 2.4 of the prev*
*ious
section.
Proof of Theorem 1.1.By [13, 2.1], a map of Ealgebras can be factored as a rel*
*ative
cell inclusion followed by an acyclic surjection. Applying this to the Ealgeb*
*ras
C*Xn, we can construct the following commutative diagram of Ealgebras.
A0_//_____//A1//___//_././._//_An//___//_. . .
  
~  ~ ~ 
fflfflfflfflfflfflfflffl fflfflfflffl
C*X0 ____//_C*X1____//._._._//C*Xn_____//. . .
~
Here as in [13] the arrows "i " denote acyclic surjections and the arrows "ae"
denote relative cell inclusions. 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 2.4 and the vertical
maps are weak equivalences by Proposition 2.5 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. __
The proof of Theorem 1.2 is similar, but needs in addition the identification
given in [13, 1.5] of the homology of the pushout of cell Ealgebras over relat*
*ive cell
inclusions in terms of the E1 torsion product of [12]. To apply this, we need *
*the
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 7
following result proved in Section 7 relating the E1 torsion product to the us*
*ual
differential torsion product.
Lemma*3.1. Let X, Y , and Z be as in Theorem 1.2. The E1 torsion product
TorC*Z(C*X;*C*Y ) is canonically isomorphic to the usual differential torsion p*
*rod
uct TorC*Z(C*X; C*Y ). Under this isomorphism, the composite
*Z * * * * * * * *
TorC* (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 [13, 1.7], we can choose cell Ealgebras A, B, C, qu*
*asi
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 3.1
and wellknown results on the EilenbergMoore map (e.g. [21, 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. __
4. 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 [12, xI.7], [17]. The theory of [17] 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
8 MICHAEL A. MANDELL
that the operations commute with "suspension" [17, 3.3], the following observat*
*ion
can be proved by the argument of [17, 8.1].
Proposition 4.1.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 5, we describe all of the Ealgebra 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 [16, p. 100]. 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 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 bel*
*ow
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
Ealgebras.
EFp [n]____//ECFp [n]
pn qn
fflffl fflffl
EFp [n]__a___//_Bn
We therefore obtain a map ff: Bn ! C*Kn. We prove the following theorem in
Section 6.
Theorem 4.2. The map ff: Bn ! C*Kn is a quasiisomorphism.
Corollary 4.3.Kn is resolvable.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 9
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
relative cell inclusion. The following two propositions then imply that UBn is*
* a
K(Z=pZ ; n).
By the theorem, the unit of the derived adjunction Kn ! UC*Kn is represented
by the map Kn ! UBn. To see that it is a weak equivalence, it suffices to see t*
*hat
the induced map on ssn is an isomorphism. The p distinct homotopy classes of ma*
*ps
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 adjunction 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 4.4.UECFp [n] is contractible.
Proof.ECFp [n] is a cell Ealgebra and so UECFp [n] and the map Fp ! ECFp [n]
is a quasiisomorphism, so the map UECFp [n] ! UFp = * is a weak equivalence_
of Kan complexes. __
Proposition 4.5.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) [16, p. 100101].
The map of simplicial sets [n] ! [n]=@[n] induces a bijection
E (EFp [n]; C*[n]) ~=E (EFp [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]):
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 4.1 and the Cartan_
formula [17, 2.7ff]. __
We complete the proof of Theorem 1.3 by deducing that K(Z^p; n) is resolvable
for n 1.
10 MICHAEL A. MANDELL
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. __
5. The Algebra of Generalized Steenrod Operations
The key to the proof of Theorem 4.2 is a study of the algebra of all generali*
*zed
Steenrod operations of [17]. Precisely, let B be the free associative Fpalgeb*
*ra
generated by the P sand (if p > 2) the fiP s[17, 2.2,x5] for all s 2 Z modulo t*
*he two
sided ideal consisting of those operations that are zero on all "Adem objects" *
*[17,
4.1] of "C (p; 1)" of [17, 2.1]. The Adem objects of C (p; 1) include all E1 al*
*gebras
over any E1 koperad for any commutative Fpalgebra k. We refer to B as the
algebra of all operations. In this section, we prove Theorem 1.4 and provide the
main results needed in the next section to prove Theorem 4.2. We use the standa*
*rd
arguments effective in studying the Steenrod and DyerLashoff algebras to analy*
*ze
the structure of B.
Definition 5.1.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 5.2.The set {P I I is admissible} is a basis of the underlying Fp
module of B.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 11
Proof.It follows from the Adem relations [17, 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 [18, 2.2 or 2.6]. _*
*_
Proposition 5.3.If s > 0 then P s(P 0)s = 0 and (if p > 2) fiP s(P 0)s = 0
(P sor fiP scomposed with s factors of P 0).
Proof.The Adem relations [17, 4.7] for fifflP sP 0when s > 0 are given by
1X
fifflP sP 0= (1)si(pi  s; s  i  1)fifflP (si)P i;
i=1
where we understand ffl = 0 when p = 2, and the notation (j; k) denotes the bin*
*omial
coefficient (j + k)!=(j!k!) when j 0 and k 0 and zero if either j < 0 or
k < 0. The coefficient (pi  s; s  i  1) therefore can only be nonzero when
s=p i s  1. Then P 1P 0= 0 and fiP 1P 0= 0 since the binomial coefficients
are zero for all values of i. Assume by induction that P t(P 0)t = 0 for all t*
* such
that 1 t < s; we see that P sP 0and fiP sP 0are both in the left ideal gener*
*ated
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 5.4.The left ideal of B generated by (1  P 0) is a twosided ideal.
Proof.By the previous proposition it suffices to show that for every admissible
sequence 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 ff*
*lj = 0 and
we think of this sequence as (s1; : :;:sk)). If sk < 0 then by what we have alr*
*eady
shown,
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 0P s= (1)i(pi; (p  1)s + i  1)P 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. *
*___
12 MICHAEL A. MANDELL
For the other half of Theorem 1.4, we need a canonical map from B to the
Steenrod algebra A. It can be shown [17, 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 4.1 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 5.5.Let k be a commutative Fp algebra and let G 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 5.6.The previous proposition and the Cartan formula [17, 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
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, observe that by what*
* we
have shown, 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_independe*
*nt._
__
6. Unstable Modules over B
In this section, we prove Theorem 4.2. The proof is based on a comparison of
free unstable modules over A with free unstable modules over B.
Definition 6.1.A module M over B is unstable if for every x 2 M of degree d,
and for every admissible sequence I with e(I) > d, P Ix = 0.
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 un
stable [17, 5.(3)(4)]. In the statement of the following proposition, the enve*
*loping
algebra of an unstable Bmodule is the free graded commutative algebra modulo
the relation that the pth power operation (the restriction) of any element is *
*its
pth power under the multiplication in the ring.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 13
Proposition 6.2.If G is an E1 Fpoperad, then H*GFp [n] is the enveloping al
gebra of the free unstable Bmodule on one generator in degree m. H*EFp [n] is
the extended Fpalgebra on the enveloping algebra of the free unstable Bmodule*
* on
one generator in degree n.
Proof.The argument of [18, 2.6] applies to prove the first statement. The secon*
*d_
statement follows from the first. __
We denote by Aunnand Bunnthe free unstable A and Bmodules on one generator
in degree n. In the following proposition, (1  P 0) denotes the map of Bmodul*
*es
Bunn! Bunnthat sends the generator to 1  P 0times the generator.
Proposition 6.3.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 5.3. 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, we now proceed by induction on n(I) to choose
f(P I) for n(I) > 0. When n(I) > 0, we can write I as the concatenation (J; (0;*
* 0))
where n(J) = n(I)  1; 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 (1P 0) sends P Ibn to P I(1P 0)bn; it clearly sends M+ into
M+ , and it follows from Propositions 5.2 and 5.3 that it sends M to M . Since
on M , f sends ffbn to ff(1 + P 0+ (P 0)2 + . .).bn, it follows that the compo*
*site
on M sends ffbn to ff(1  P 0)(1 + P 0+ (P 0)2 + . .).bn = ffbn. To see that t*
*he
composite is the identity on M+ , it suffices to check it on a standard basis e*
*lement,
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. _*
*__
14 MICHAEL A. MANDELL
Proof of Theorem 4.2.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)
discussed above by In; by Proposition 6.3, Bunnis isomorphic as a restricted Fp
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
TorVIn*;*(Fp ; VBunn) ~=Fp VInVBunn~=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 6.2 identifies H*EFp [n] as VBunn. It is well
known that H*Kn = H*K(Z=pZ ; n) can be identified with VAunn. We see from
Proposition 5.5 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 (1P 0): Bunn! Bunn; in other words, we can identify the map induced by pn *
*on
homology as the inclusion VIn ! VBunn. By [13, 1.5], the spectral sequence of [*
*12,
V.7.3] calculates the homology groups of the pushout Bn. This spectral sequence
has E2 term TorVIn*;*(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. __
7.The E1 Torsion Product and the EilenbergMoore Map
In this section we prove Lemma 3.1. The proof consists of an adaptation of the
results of [13] to compare a bar construction in the category of Ealgebras to *
*the
cochain complex of the cobar construction of spaces.
Recall that for maps of simplicial sets X ! Z and Y ! Z, the cobar constructi*
*on
Cobaro(X; Z; Y ) is the cosimplicial simplicial set that is given in cosimplici*
*al 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
*Z * * * * o
TorC (C X; C Y ) ! H (N(C (Cobar (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
*Z * * * * o *
TorC (C X; C Y ) ! H (N(C (Cobar (X; Z; Y )))) ! H (X xZ Y )
is the EilenbergMoore map.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 15
The corresponding construction in the category of Ealgebras is the bar const*
*ruc
tion. For 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 Fpmodules N(fio(B; A; C)) ! B qA C. According to [13,
1.6], when A is a cell Ealgebra, and A ! B and A ! C are relative cell inclusi*
*ons,
the natural map
N(fio(B; A; C)) ! B qA C
is a quasiisomorphism.
The proof of Lemma 4.2 is a straightforward comparison of these two construc
tions.
Proof of Lemma 4.2.Using [13, 1.7], we can find cell Ealgebras A, B, C, relati*
*ve
cell inclusions A ! B, A ! C, and quasiisomorphisms A ! Z, B ! X, C ! Y ,
such that the following diagram commutes.
B oo_____oAo_//____//__C
  
~  ~  ~
fflfflfflfflfflfflfflfflfflfflfflffl
C*X oo___C*Z _____//C*Y
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 ):
By [13, 1.4] and the K"unneth theorem, the composite above is a quasiisomorphi*
*sm.
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 )
*Z
By [13, 1.5], H*(B qA C) is the E1 torsion product TorC (C*X; C*Y ), and un
der this*identification, the map B qA C ! C*X qC*Z C*Y is the canonical_map_
TorC Z(C*X; C*Y ) ! C*X qC*Z C*Y . The lemma now follows. __
16 MICHAEL A. MANDELL
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 an arbitrary field k, there is no difficulty in providing a natural Ekal*
*gebra
structure on the cochains of simplicial sets, for some E1 koperad Ek. For exa*
*mple
the work of [10] and the construction described in Section 1 produce such a str*
*uc
ture. Write Ek for the category of Ekalgebras. 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 2, we obtain the following proposition.
Proposition A.2. 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
H*(; k)local nilpotent simplicial set of finite ktype is kresolvable. The *
*base
field Fp is irrelevant in Sections 3 and 7, and the arguments there apply to pr*
*ove
the following propositions that allow us to argue inductively up principally re*
*fined
Postnikov towers.
Proposition A.3. 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.4. 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 onl*
*y 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.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 17
By the argument in Section 4, 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.5. 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.6. 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 4 and
prove that the map ffk: Bn ! C*(Kn; k) is a quasiisomorphism just as in Sec
tion 6. 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 4.5 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. __
Appendix B. 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 [6] comparing the padic
homotopy category with the homotopy category of E1 HFp ring spectra.
Theorem B.1. (DwyerHopkins) The free mapping spectrum functor F (()+ ; Fp)
induces an equivalence between the homotopy category of connected nilpotent spa*
*ces
of finite ptype and a full subcategory of the homotopy category of E1 HFp ri*
*ng
spectra.
By the homotopy category of E1 HFp ring spectra, we mean the category ob
tained from the category of E1 ring spectra under the (cofibrant) E1 ring spe*
*c
trum 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
18 MICHAEL A. MANDELL
ordinary cohomology with coefficients in Fp in the sense that there is a canoni*
*cal
map H*(X; Fp) ! ss*(F (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 [8]. 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, a*
*nd
so it suffices to prove that the functor F induces an equivalence between the h*
*omo
topy category of connected nilpotent spaces of finite ptype and a full subcate*
*gory
of the homotopy category of commutative HFp algebras. We denote the category
of commutative HFp algebras as C . By [8, VII.4.10], C is a closed model categ*
*ory
with weak equivalences the weak equivalences of the underlying spectra; we deno*
*te
its homotopy as hC .
The commutative HFp algebra F X is the "cotensor" of HFp with X [8, 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
unique up to canonical isomorphism, and [8, VII.2.9] guarantees that they exist*
* for
any A and any X. The significance of the identification of F X as the tensor is*
* in
the following proposition.
Proposition B.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 [8, VII.4.1*
*6]
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 B.3. The functor T preserves weak equivalences between cofibrant
objects.
As a slight generalization of the proof of [8, VII.4.16], it is elementary to*
* 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. Applying T and using the tensor adjunction*
*, we
obtain the following proposition.
Proposition B.4. The functor T converts cofibrations to fibrations.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 19
The functors F and T are therefore a model category adjunction. In particular,
we obtain the following proposition.
Proposition B.5. The (right) derived functors F and T exist and give a con
travariant right adjunction H o(X; T A) ~=hC (A; F X).
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 ! T F X is a weak equivalence. Thus, we ne*
*ed
to show that if X is a connected nilpotent pcomplete 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 B.4
by essentially the same argument used to prove Theorem 1.1 from Proposition 2.4.
Proposition B.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 B.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 the this theorem is essentially the same in outline as the proof*
* of
Theorem 1.2. The analogue of Lemma 3.1 can be proved by observing that the bar
construction of the cofibrant approximations in C is equivalent to the (thicken*
*ed)
realization of F applied to the cobar construction of the singular simplicial s*
*ets on
the spaces X, Y , and Z. Some fiddling with the filtration induced by the cosim
plicial direction of the cobar construction and the filtration induced by the s*
*keletal
filtration*of the singular simplicial sets allows the identification of TorFZ**
*(F X; F Y )
as TorC*Z(C*X; C*Y ) and the composite map
*Z * * FZ *
TorC* (C X; C Y ) ~=Tor*(F X; F Y ) ! ss*F (X xZ Y ) ~=H (X xZ Y )
as the EilenbergMoore map.
To complete the proof of Theorem B.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 nilpotent pcomple*
*te
space of finite ptype is HFp resolvable. The remainder of the appendix is dev*
*oted
to sketching a proof of the following theorem.
Theorem B.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 the "free" commutative HFp algebra
on the spectrum Sn , denoted PSnHFpin [8] is 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
20 MICHAEL A. MANDELL
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 B.9. For n 1, the map Bn ! F K(Z=pZ ; n) is a weak equivalence.
The proof uses the EilenbergMoore spectral sequence of [8, IV.4.1] in place *
*of
the EilenbergMoore spectral sequence of [12, V.7.3], but otherwise is the same*
* as
the proof of Theorem 4.2.
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 B.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 4.3 shows that the map
K(Z=pZ ; n) ! T Bn is a weak equivalence, completing the proof of Theorem B.8.
Unfortunately, the simple algebraic argument given in Proposition 4.5 to iden*
*tify
ffn as 1  in the algebraic case does not have a topological analogue. Here we
must use the topology 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 B.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 [17, 3.3].
Proposition B.11. The suspension homomorphism oe commutes with the opera
tion P sfor all s.
E1 ALGEBRAS AND pADIC HOMOTOPY THEORY 21
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 B.10.
It follows from Proposition B.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  .
References
[1]A. K. Bousfield, "The Localization of Spaces with Respect to Homology," Top*
*ology 14 (1975),
pp. 133150.
[2]A. K. Bousfield, "On the Homology Spectral Sequence of a Cosimplicial Space*
*," Amer. J.
Math. 109 (1987), pp. 361394.
[3]A. K. Bousfield, V. K. A. M. Gugenheim, On PL De Rham Theory and Rational H*
*omotopy
Type, Memoirs Amer. Math. Soc. 179, 1976.
[4]W. G. Dwyer, "Strong Convergence of the EilenbergMoore Spectral Sequence,"*
* Topology 13
(1974), pp. 255265.
[5]W. G. Dwyer, "Exotic Convergence of the EilenbergMoore Spectral Sequence,"*
* Illinois J.
Math. 19 (1975), pp. 607617.
[6]W. G. Dwyer, M. J. Hopkins, unpublished work, 1992.
[7]W. G. Dwyer, J. Spalinski, "Homotopy Theories and Model Categories," in I. *
*M. James, ed,
Handbook of Algebraic Topology, Elsevier Science B.V., 1995.
[8]A. D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May, Rings, Modules, and Alg*
*ebras in
Stable Homotopy Theory, Amer. Math. Soc. Surveys & Monographs, vol. 47, 1996*
*. Errata
http://www.math.uchicago.edu/"mandell/ekmmerr.dvi.
[9]P. G. Goerss, "Simplicial Chains over a Field and pLocal Homotopy Theory,"*
* Math. Z. 220
#4 (1995), pp. 523544.
[10]V. A. Hinich, V. V. Schechtman, "On Homotopy Limit of Homotopy Algebras," K*
*Theory,
Arithmetic, and Geometry, Springer Lecture Notes 1289 (1987), pp. 240264.
[11]I. Kriz, "pAdic Homotopy Theory," Top. & Appl. 52 (1993), pp. 279308.
[12]I. Kriz, J. P. May, Operads, Algebras, Modules, and Motives, Asterisque 233*
*, 1995.
[13]M. A. Mandell, "The Homotopy Theory of E1 Algebras," submitted in conjuncti*
*on with
this manuscript.
[14]M. A. Mandell, "Algebraization of E1 Ring Spectra," in preparation.
[15]M. A. Mandell, "Natural E1 Multiplications on Cochain Complexes," in prepar*
*tion.
[16]J. P. May, Simplicial Objects in Algebraic Topology, Van Nostrand, 1967.
[17]J. P. May, "A General Algebraic Approach to Steenrod Operations," Steenrod *
*Algebra and
its Applications: A Conference to Celebrate N. E. Steenrod's Sixtieth Birthd*
*ay, Springer
Lecture Notes 168 (1970), pp. 153231.
[18]J. E. McClure, "The mod p KTheory of QX," H1 Ring Spectra and their Applic*
*ations,
Springer Lecture Notes 1176 (1986), pp. 291377.
[19]D. G. Quillen, Homotopical Algebra, Springer Lecture Notes 43, 1967.
[20]V. A. Smirnov, "Homotopy Theory of Coalgebras," Izv. Akad. Nauk SSSR Ser. M*
*at. 49
(1985), pp. 13021321, trans. in Math. USSRIzv. 27 (1986), pp. 575592.
[21]L. Smith, "Homological Algebra and the EilenbergMoore Spectral Sequence," *
*Trans. Amer.
Math. Soc. 129 (1967), pp. 5893.
[22]D. Sullivan, "Infinitesimal Computations in Topology," Publ. Math. I. H. E.*
* S. 47 (1978), pp.
269331.
Department of Mathematics, University of Chicago, Chicago, IL
Current address: Department of Mathematics, M. I. T., Cambridge, MA
Email address: mandell@math.mit.edu