baselineskip 16pt .
Fibrewise Localization
and
Unstable Adams Spectral Sequences
by
William Dwyer
Haynes Miller
and Joseph Neisendorfer
Typeset by AM S-TEX
0. Introduction.
Let LX be localization with respect to some homology theory and
let X ! LX be the localization map. If E ! B is a fibration, then
fibrewise localization KE is defined to be a homotopy pullback
KE ! LE
# #
B ! LB :
This definition of fibrewise localization is completely general but, to be
useful, it should have three properties.
First, we do not know in general that LE ! LB has LF as its homo-
topy theoretic fibre, where F is the fibre of E ! B. Hence, we do not
know that the homotopy theortic fibre of KE ! B is LF and this, of
course, should be true of anything called fibrewise localization. Suppose
L is localization with respect to homology with coefficients in a ring R
where R is either Z=nZ or a subring of Q. Then a satisfactory solution
is given by, among others, Bousfield-Kan [2]. They construct a natural
transformation X ! R1 X such that, at least for nilpotent X, R1 X is
the localization of X with respect to H( ; R). In addition, if ss1B acts
nilpotently on Hi(F ; R) for all i, then the homotopy theoretic fibre of
R1 E ! R1 B is R1 F . For example, this is the case if p is a prime, ss1
is a p-group, and R = Fp = the field with p elements.
Second, if Y is a space over B, we would like to get some hold
on the homotopy groups of the space (Y; KE) of maps Y ! KE
over B. The Bousfield-Kan construction of Fp1 X gives an unstable
Adams spectral sequence for computing the homotopy groups of the s-
pace map*(Y; Fp1 X) of pointed maps. We are led to imitate this in
order to compute the homotopy groups of (Y; KE). Furthermore, the
E2 term of the Bousfield-Kan unstable Adams spectral sequence can be
identified by nonabelian homological algebra. Bousfield-Kan do give a
construction of fibrewise localization, denoted F_p1 E, and their construc-
tion gives an unstable Adams spectral sequence. But their construction
does not seem to permit an algebraic description of the E2 term. We
give in this paper a new construction of fibrewise localization, denot-
ed BR1 (E), which gives when R = Fp a spectral sequence with an
algebraically identifiable E2 term.
Third, this spectral sequence gives information about (Y; BFp1 (E))
but most often one really wants to know about (Y; E). We have not
resolved the question of what are the properties of the natural maps
2
(Y; E) ! (Y; BFp1 (E)) but a special case has been settled by the
first author. Suppose ss is a p-group, X is a simply connected space
with a ss-action, and E ! B is the Borel construction Ess xssX !
Bss. Then, if Y ! B is the identity Bss ! Bss, the map (Bss; E) !
(Bss; BFp1 (E)) is a mod p homotopy isomorphism, that is, the fibres
are simple spaces with homotopy groups which are uniquely p-divisible,
[3].
We apply our unstable Adams spectral sequence to give another proof
of the generalized Sullivan conjecture. This proof follows closely a proof
by Lannes [4] using the Bousfield-Kan Adams spectral sequence.
We would like to express our gratitude to A.K. Bousfield for his many
letters to us describing unstable Adams spectral sequences, bicosimpli-
cial spaces, the role of derivations, and convergence criteria.
Throughout this paper, our basic reference is Bousfield-Kan [2]. Many
of the ideas there are also expounded in [5].
3
1. A construction of fibrewise localization.
We work in the category of simplicial sets and adopt the convention
that "space" means "simplicial set." The Bousfield-Kan construction of
the localization R1 X is done in two stages. In the first stage, a triple
denoted R is used to construct a canonical resolution X ! RoX which
is an augmented cosimplicial space. In the second stage, a functor tot
from cosimplicial spaces to spaces is applied to Xo . The result tot Xo
is R1 X. Since our construction of fibrewise localization fits into this
framework, we describe this process in more detail.
Let T be a triple with triple structure j : I ! T and : T 2 ! T .
Given an object X, the canonical resolution of X is the augmented
cosimplicial object X ! T oX defined by:
(T oX)s = T s+1X
di = T ij(T s-iX) : (T oX)s-1 ! (T oX)s
si = T i(T s-iX) : (T oX)s+1 ! (T oX)s
where i = 0; : :;:s
and the augmentation is d0 = j : X ! (T oX)0.
It is common to write Ao; Bo ; Co ; : : :for cosimplicial objects and
A-1 ! Ao or X ! Ao if Ao is augmented.
It is useful to be aware of the following equivalent formulation *
*of
an augmented cosimplicial object X ! Ao. Write X__ for the constant
cosimplicial object, i.e. X__s= X for all s 0 and all coface operators
di and codegeneracy operators si are identity maps. An augmentation
X ! Ao is equivalent to a cosimplicial map X__! Ao.
The first step of the Bousfield-Kan construction of R1 X is the canon-
ical resolution X ! RoX associated to the triple R defined as follows.
If X is a space and R is a commutative ring, then RX is the space
of finite formal sums r[x] where r is in R and x is in X. The triple
structure j(X) : X ! RX and (X) : R2X ! RX is: j(X) = [x] and
(s[r[x]]) = sr[x]. The second stage of the Bousfield-Kan construc-
tion is to assign to any cosimplicial space Y o a space tot Y o and then
R1 X is tot RoX. The functor tot is described as follows:
Let o be the cosimplicial space with (o)s = s and with di and si
the standard coface and codegeneracy maps. Given cosimplicial spaces
Ao and Bo , let hom (Ao; Bo ) be the cosimplicial mapping space where
4
the n-simplices hom (Ao; Bo )n are the set of cosimplicial morphisms n x
Ao ! Bo . Here, (n x Ao)s = n x As, di = 1 x di, and si = 1 x si.
The face operators di : hom (Ao; Bo )n ! hom (A; B)n-1 and degeneracy
operators si : hom (Ao; Bo )n ! hom (Ao; Bo )n+1 are di(f ) = f (di x 1)
and si(f ) = f (si x 1). Let tot Y o = hom (o; Y o).
It is not difficult to check that, if X__ is constant, then tot X__ = X.
Hence, if X ! Y o is an augmented cosimplicial space, applying tot to
X__! Y o gives a map X ! tot Y o.
If T is a triple defined on any category which admits a forgetful functor
to spaces, then applying tot to the canonical resolution X ! T oX gives
a map of spaces X ! tot T oX = T1 X.
We construct fibrewise localization by applying this procedure to the
following triple BR defined on the category of spaces over a fixed space
B.
Given a space E over B with map p : E ! B, let BR(E) = B x RE
and regard BR(E) as a space over B by projecting on the first factor.
The triple structure j : 1 ! BR and : (BR)2 ! BR is: j(E) =
(p; j(E)) : E ! BxRE and (E) = 1x((E)R(ss2)) : BxR(BxRE) !
B x RE where ss2 : B x RE is projection on the second factor.
Our candidate for fibrewise localization is the resulting space BR1 (E) =
tot BRo(E). Since there is a map BRo(E) ! B__, BR1 (E) is a space
over B and E ! BR1 (E) is a map over B.
If E and B are fibrant, i.e. Kan complexes, we show in section 7 that
BR1 (E) is a homotopy pullback of
R1 E
# :
B ! R1 B
Hence, BR1 (E) is a fibrewise localization of E.
5
2. Derived functors of derivations.
Let CA__ denote the category of graded commutative unstable coalge-
bras with unit over the mod p Steenrod algebra. If B is an object in
CA__, let CA__=B be the category of objects over B.
Let St = H*(St; Fp) in CA__ and let X be an object in CA__=B. The
projection StX ! X makes StX into an object in CA__=B so that this
projection is a map in CA__=B. If t > 0, there is a unique augmentation
Fp ! St. If t = 0, pick one. This gives a map X ! St X.
If : X ! E is a map in CA__=B, then a map ' : St X ! E in
CA__=B is called a derivation of degree t with respect to if
X - ! E
# % '
St X
___
is commutative. Of course, ' is determined by its restriction to H n (St; Fp)
X. Hence, when t = 0, a derivation is just a map in CA__=B and, when
t 1, a derivation is just the usual definition of a coalgebra derivation
over the Steenrod algebra. Thus, if DertCA_=B(X; E) is the set of all such
derivations, then Der 0CA_=B(X; E) = Hom CA__=B(X; E) and, if t 1,
Der tCA_=B(X; E) is an Fp module.
Let F__pbe the category of positively graded Fp modules. The forgetful
functor J : CA__ ! F__phas a right adjoint G : F__p! CA__. We also write
G for the resulting triple G = GJ : CA__ ! CA__ with triple structure
j : I ! G and : G2 ! G. Now define a triple B - G : CA__=B ! CA__=B
as follows: If E is in CA__=B with map p : E ! B, let (B - G)(E) =
B G(E), j(E) = (p j) : E ! E E ! B G(E), and (E) =
(1)(1G(ss2)) : B G(B G(E)) ! B G(G(E)) ! B G(E) where
ss2 : B G(E) ! G(E) is the projection given by the unit B ! Fp.
If Ao is any cosimplicial group, the cohomotopy group sssAo is the
cohomology group Hs (Ao; d) where d = (-1)idi. If Ao is a cosimplicial
set, we may still define ss0Ao as the coequalizer of d0 and d1 : A0 ! A1.
If : X ! E is a map in CA__=B, then the augmentation E !
(B - G)o(E) gives unique maps X ! (BG)n+1 (E) which we also denote
by . The right derived functors of Der tCA_=B(X; E) are defined, for
all s 0 if t 1 and for s = 0 if t = 0, by Ext s;tCA_=B(X; E) =
sss(Der tCA_=B(X; (B - G)o(E)) ).
6
Since coalgebras have units ffl : E ! Fp, CA__ = CA__=Fp and we have
[4].
Lemma 2.1. Let E be an object in CA__ with augmentation j : Fp ! E.
If E is connected, then Ext s;tCA_(Fp; E)j = 0 for t - s 0 unless s = t =
0.
If B and F are objects in CA__, let B F be the object in CA__=B with
map ss1 = 1 ffl : B F ! B. Given a map : X ! B F in CA__=B,
it is easy to verify that:
Lemma 2.2. Der CA__=B(X; B F ) = Der sCA_(X; F )ss2 where ss2 = ffl1.
More generally, Ext s;tCA_=B(X; B F ) = Ext s;tCA_(X; F )ss2 .
All of the preceding dualizes to algebras as follows: Let A_ be the
category of unstable algebras over the mod p Steenrod algebras. If B*
is an object in A_, let A_ \ B* be the category of objects under B* , that
is, of objects E* in A_ together with maps p* : B* ! E* . The forgetful
functor J : A_ ! F__phas a left adjoint G* and we also write G* for the
cotriple G* = G*J. There is a cotriple (B* - G*) on A_ \ B defined by
(B* - G*)(E* ) = B* G*(E* ). If * : E* ! X* is a map in A_ \ B* ,
we define Der A_\B* E* ; X* ) * and Ext s;tA_\B*(E* ; X* ) * by the obvious
dual procedure. Then, if X; E; B are finite type objects in CA__ with dual
objects X* ; E* ; B* in A_, we have
Der sCA_=B(X; E) = Der sA_\B*(E* ; X* ) * and
Ext s;tCA_=B(X; E) = Ext s;tA_\B*(E* ; X* ) * :
If ffl : E* ! Fp is an augmented object in A_, let E*c= F pE0 E* be
the connected component with induced augmentation __ffl: E*c! Fp.
Lemma 2.3. If p* : B* ! E* is a map of augmented objects, then
Der A_\B* (E* ; Fp)ffl= Der sA_\B*c(E*c; Fp)_ffland Ext s;tA_\B*(E* ; Fp)ffl=*
* Ext s;tA_\B*c(E*c; Fp)_ffl.
The proof of the above is essentially the same as the proof for the
special case A_ \ B* = A_ in [4].
Let ss = Z=pZ and H = H* Bss. Recall that Lannes [4] has given a
functor T : A_ ! A_ defined as a left adjoint, Hom A_(B* ; H C* ) =
Hom A_(T B* ; C* ). It has the following remarkable properties:
a) T preserves free objects,
7
b) T preserves tensor products, T (A B) = T A T B,
c) T is exact, that is, T preserves simplicial resolutions, and
d) if A is finite dimensional, the natural map Fp ! H induces an
isomorphism T H ! H.
From these properties, it follows easily that e) T preserves derivations,
Der sA_\B*(E* ; H X* ) * = Der sA_\T B*(T E* ; X* ) * where __* : T E* !
X* is the morphism corresponding to * : E* ! H X* , and f) T pre-
serves derived functors, Ext s;tA_\B*(E* ; HX* ) * = Ext s;tA_\T(B*T E* ; *
*X* )__*.
8
3. An unstable Adams spectral sequence.
In this section, we describe in our context the extended homotopy
spectral sequence of Bousfield-Kan and a variation of it due to Bousfield.
If W is a space, let W [s]denote its s skeleton. If Xo is a cosimplicial
space, let totsXo = hom (o[s]; Xo ) for s 0 and let tot-1 Xo = a point.
Then tot0Xo = X0 and the natural maps totsXo ! tot s-1 Xo form
an inverse system with lim totsXo = tot1 Xo = totXo .
If Xo is pointed, letT N oXo be the normalized object defined by
N 0Xo = X0 and N nXo = n-1i=0kersi : Xn ! Xn-1 for n 1. The
fibre of totsXo ! tots-1 Xo is the pointed mapping space sN sXo =
map*(Ss; N sXo ).
If Xo is a cosimplicial group, then the differential d = (-1)idi acts
on N oXo with Hs (N oXo ; d) isomorphic to sssXo .
Let : Y ! E be a map of spaces over B and let (Y; E) be the space
of all such maps with as the basepoint. If E ! Xo is an augmented
cosimplicial space over B, then defines unique maps Y ! Xn for
n 0 and hence a map Y ! tot Xo . These maps are also denoted
by . Hence, (Y; Xn ), (X; tot Xo ), and (Y; E) ! (Y; Xo ) are all
pointed.
Exercise. tot (Y; Xo ) = (Y; tot Xo ).
In particular, tot (Y; BRo(E)) = (Y; BR1 (E)). Furthermore, s-
ince (Y; BRo(E)) is grouplike [2], these spaces are fibrant.
Write tots for tots(Y; BRo(E)). The tower of maps tots ! tots-1
is a tower of R-principal fibrations which are not necessarily surjec-
tive. Since lim tot s = (Y; BR1 (E)), the exact homotopy sequences
of these fibrations fit together, as in Bousfield-Kan , into an unstable
Adams spectral sequence. That is, Es;tr= Es;tr(Y; E; ) is defined for
t s 0 and there are differentials dr : Es;tr! Es+r;t+r-1r . The
abutment is sst-s (Y; BR1 (E)) and Es;t1= sst-s (sN s(Y; BRo(E)) =
sst N s(Y; BRo(E)) = N ssst (Y; BRo(E)). Furthermore, d1 is d =
(-1)idi and therefore Es;t2= ssssst (Y; BRo(E)).
The following convergence results follow from Bousfield-Kan . If k 0
and Es;t2(Y; E; ) = 0 for 0 t - s k, then (Y; BR1 (E)) is k-
connected. If : E ! E0 is a map over B, Es;t2(Y; E; ) = 0 for t-s = 0,
and Es;t2(Y; E; ) ! Es;t2(Y; E0 ) is an isomorphism for t - s 0, then
9
(Y; BR1 (E)) ! (Y; BR1 (E0)) is a homotopy equivalence.
Now, let R = Fp and H*X = H*(X; Fp). The topological triple BFp
and the algebraic triple H*(B) - G are related by a natural isomor-
phism : H*(BFp(E)) ! (H*B - G)(H*E) which respects the triple
structures. That is, the following diagrams commute.
H*E H*(j)-! H*(BFp(E))
#
j &
(H*B - G)(H*E)
H*(BFp2(E)) H*()-! H*(BFp(E))
(H*B - G)(H*(BFp(E))) #
(H*B - G)()
(H*B - G)2(H*E) -! (H*B - G)(H*E)
Regarding St ! (Y; E) as a map StxY ! E gives a map sst(Y; E) !
Der tCA_=H*B (H*Y; H*E)H* which is an isomorphism when E = BFp(E0) =
B x Fp(E0).
Hence, Es;t1(Y; E; ) = N ssst(Y; BFpo(E)) = N sDer tCA_=H*B(H*Y; H*BFpo(E))*
*H* =
N sDer tCA_=H*B(H*Y; (H*B-G)o(H*E))H* and Es;t2(Y; E; ) = Ext s;tCA_=H*B(H*
**Y; H*E)H* .
Bousfield points out that the following variation occurs when one con-
siders only the component (Y; BFp1 (E)) of . One gets a spectral
sequence Es;tr(Y; E) with Es;t2(Y; E) = Ext s;tCA_=A*B(H*Y; H*E)H*
for 1 t s 0 and = 0 otherwise. If lim 1Es;tr(Y; E) = 0 for all
s 0 and t > s, this spectral sequence satisfies the following strong
convergence condition: When t > s, sst-s (Y; BFp1 (E)) is the inverse
limit of Qs = image of sst-s (Y; BFp1 (E)) ! sst-s tots. The kernels
es;t of the maps Qs ! Qs-1 are Es;t1(Y; E) .
When t = s, let "ss0= the set of ' in ss0(Y; E) such that H*' = H* .
Then the map "ss0! lim Qs is a surjection with trivial kernel and the
kernel es;s injects into Es;s1(Y; E) for s 0.
Bousfield also proves the following result. Let ff : H*Y ! H*E be a
map in CA__=H*B such that Ext s;s-1CA_=H*B(H*Y; H*E)ff= 0 for all s 2.
10
Then there exists a map ' : Y ! BR1 (E) over B with H*' = ff.
Moreover, the canonical injections es;s ! Es;t1(Y; E) are bijections. In
particular, if Ext s;sCA_=H*B(H*Y; H*E)ff= 0 for s 0, then ' is unique
up to homotopy over B.
Now, we give an application. Let ss = Z=pZ, X = a finite CW complex
with a cellular ss-action, and Xss = the fixed point set. Let Ess(X) =
Ess xssX be the Borel construction and note Ess(Xss) = Bssx Xss !
Ess(X). As usual, spaces may be identified with their singular complexes,
so that we may apply BRp1 and BFp1 (Ess(X) = Ess(Fp1 X).
Proposition 3.1 (Generalized Sullivan conjecture). There is a
weak homotopy equivalence from Fp1 (Xss) to the space (Ess(Fp1 X))
of sections of Ess(Fp1 X).
Proof: Since Xss is finite dimensional, the Sullivan conjecture [5] im-
plies that (Ess(Fp1 Xss)) = map(Bss; Fp1 Xss) ' Fp1 X. Hence, 3.1
reduces to showing that (Ess(Fp1 Xss)) ! (Ess(Fp1 X)) is a weak e-
quivalence of spaces of sections. Letting Bss ! Bss be the identity, we
must show that
(Bss; Ess(Fp1 Xss)) ! (Bss; Ess(Fp1 X))
is a weak equivalence.
`
If H = H* Bss, a basic computation of Lannes [4] is: T H = 'ffl hom(ss;ss)H
with 1 : H ! H corresponding to the projection on the degree 0 com-
ponent of the ' = 1 component, T H ! H ! Fp.iAnother computation
` j `
of Lannes [4] is: if E = Ess(X); T H* E = '6=0 H H* Xss H* E
with the map T H ! T H* E being given as the natural inclusion H !
H H* Xss on the ' 6= 0 components and as p* : H ! H* E on the
' = 0 component.
For any homomorphism * : H H* Xss ! H under H, Ext s;tA_\H(H
H* Xss; H) * = Ext s;tA_(H* Xss; H) * = Ext s;tA_(T H* Xss; Fp)_*__= Ext s;tA*
*_(H* Xss; Fp)_*__=
Ext s;tA_(H* Xssc; Fp)_____*, where : H* Xss ! H H* Xss is the natural in-
clusion. In particular, 2.1 implies that this = 0 for t - s 0 unless
s = t = 0.
For any homomorphism '* : H* E ! H under H, Ext s;tA_\H(H* E; H)'* =
Ext s;tA_\T(HT H* E; Fp)'_* = Ext s;tA_\T(HcT H* Ec; Fp)__'_*= Ext s;tA_\H(HH* *
*Xss; Fp)__'_*=
11
Ext s;tA_(H* Xss; Fp)__'_*. Hence, if '* = * j* : H* E ! H H* Xss !H,
then Ext s;tA_\H(H H* Xss; H) * ! Ext s;tA_\H(H* E; H)'* is an isomorphis-
m.
Setting s = t = 0 gives that hom A_\H(HH* Xss; H) ! hom A_\H(H* E; H)
is an isomorphism and, by Bousfield, ss0(Bss; BssxXss) = ss0(Bss; Bssx
Fp1 X) ! ss0(Bss; Ess(Fp1 X)) is a bijection. Since the unstable Adams
spectral sequences for each component map isomorphically at E2, (Bss; Bssx
Fp1 X) ! (Bss; Ess(Fp1 X)) is a weak equivalence. ||
12
4. Fibrations and weak equivalences.
If Xo is a cosimplicial space, then Bousfield-Kan defines for n -1
the matching spaces M nXo as follows: If n = -1, M nXo is a point *
and if n 0, M nXo is the subset of the n+1 fold product Xn x. . .xXn
consisting of those (xo; : :;:xn ) such that sixj = sj-1 xi whenever 0
i j < n. If n -1, there are natural maps
(*)n : Xn+1 ! M nXo
given by x 7! (s0x; : :;:sn x) if n 0.
These matching spaces are used by Bousfield-Kan to define fibrations
in the category of cosimplicial spaces. A map f : Xo ! Y o is called a
fibration if, for all n -1, the natural maps into the fibre product
(**)n : Xn+1 ! Y n+1 xMn Y o M nXo
are fibrations. A cosimplicial space Xo is called fibrant if Xo ! *_is a
fibration, equivalently, if the maps (*)n are fibrations for all n -1.
It is useful to note that these definitions of fibration and of fibrant
object do not depend on the coface operators.
Fibrations have the following properties. Products of fibrations are
fibrations and hence products of fibrant cosimplicial spaces are fibrant.
Pullbacks of fibrations are fibrations and hence fibres of fibrations are
fibrant. The more difficult proposition below is proved by Bousfield-Kan.
Proposition 4.1. If Eo ! Bo is a fibration with fibre F o, then tot
Eo ! tot Bo is a fibration with fibre tot F o. In particular, if Xo is
fibrant, then tot Xo is a fibrant space, that is, a Kan complex.
We now list seven useful examples.
0) Pullbacks of fibrations are fibrations.
1) If B is a fibrant space, then the constant cosimplicial space B__is
fibrant. The maps (*)n are B ! * if n = -1 and isomorphisms
if n 0.
2) Any isomorphism Xo ! Y o is a fibration.
3) If Y o is fibrant, then the projection Xo x Y o ! Xo is a fibration.
4) A cosimplicial space Xo is called grouplike if, for all n 0, Xn is
a simplicial group and the coface and codegeneracy operators ex-
cept possibly for d0 are all homomorphisms. Bousfield-Kan show
13
that grouplike objects are fibrant. For example, the canonical
resolutions RoY are grouplike. More generally, they show that
any surjective homomorphism of grouplike objects is a fibration.
5) Let p : E ! B be a space over B with E and B fibrant. Consider
the triple (B x -)(E) = B x E where B x E is regarded as a
space over B by projecting on the first factor and where the triple
structure j(E) : E ! B x E and (E) : B x B x E ! B x E
is given by j(E) = (p; 1) and (E) = projection on the first
and third factors. The canonical resolution E ! (B x -)o(E) is
called the Rector complex. The natural map (B x -)o(E) ! B__
is a fibration since the map (**)n is B x E ! B if n = -1, and
an isomorphism if n 1.
6) Let p : E ! B be a space over B and consider the resolution E !
BRo(E) introduced in section 1. We claim that q : BRo(E) ! B__
is a fibration. Pick a basepoint in B and let F o be the fibre of
q. If we forget the coface operators, then BRo(E) ! B__is the
projection B__x F o ! B__. Since F o is grouplike, F o is fibrant.
Hence, BRo(E) ! B__is a fibration. In particular, if B is fibrant,
then so is BRo(E).
Bousfield-Kan call a map f : Xo ! Y o a weak equivalence if f n :
Xn ! Y n is a weak equivalence for all n 0. The following result of
Bousfield-Kan relates this notion to fibrant objects.
Proposition 4.2. If f : Xo ! Y o is a weak equivalence with Xo and
Y o fibrant, then tot f : tot Xo ! tot Y o is a homotopy equivalence.
14
5. Bicosimplicial spaces.
If Aoo is a bicosimplicial object, then denote the horizontal operators
by dih: An-1;m ! An;m and sih: An+1;m ! An;m and the vertical op-
erators by div: An;m-1 ! An;m and siv: An;m+1 ! An;m . For example,
if Xo and Y o are cosimplicial spaces, then Xo ^xY o is the bicosimplicial
space with (Xo ^xY o)n;m = Xn x Y m and dih = di x 1, sih = si x 1,
div= 1 x di, siv= 1 x si.
If Aoo and Boo are two bicosimplicial spaces, then define the bicosim-
plicial mapping space hom (Aoo; Boo ) to be the space with n-simplices
hom (Aoo; Boo )n equal to the set of bicosimplicial maps n x Aoo ! Boo
and with face and degeneracy operators di(f ) = f (di x 1) and si(f ) =
f (si x 1):
The following lemma, which relates bicosimplicial and cosimplicial
mapping spaces, is left as an exercise.
Lemma 5.1. If Xo and Y o are cosimplicial spaces and Coo is a bi-
cosimplicial space, then hom (Xo ^xY o; Coo ) = hom (Xo ; hom (Y o; Coo ))
where hom (Y o; Coo )n is the cosimplicial space with hom (Y o; Coo )n =
hom (Y o; Cno ).
For any bicosimplicial space Aoo, define tot Aoo to be the space
hom (ox^o; Aoo). Also, define horizontal and vertical tot functors
by: toth Aoo is the cosimplicial space with (tot hAoo)m = totAom and
totv Aoo is the cosimplicial space with ( totv Aoo)n = tot Ano . The
preceding lemma has the following corollary.
Corollary 5.2. If Aoo is a bicosimplicial space, then tot Aoo = tot tothAo*
*o =
tot totvAoo.
Note. One also has tot Aoo = tot ( diag Aoo) where diag Aoo is the
cosimplicial space with (diag Aoo)n = Ann and with coface and code-
generacy operators, di = dihdiv= divdih, si = sihsiv= sivsih.
If Aoo is a bicosimplicial space, we define horizontal and vertical cosim-
plicial mapping spaces by (MhnAoo)m = M nAom and (MvmAoo)n =
M mAno .
Definition 5.3. A map g : Aoo ! Boo of bicosimplicial spaces is a
fibration if An+1;o ! Bn+1;o xMnhBoo MhnAoo is a fibration of cosimplicial
spaces for all n -1.
15
Remark. The above definition is due to Bousfield. He points out that
this is part of the definition of a closed model category in which a map
g : Aoo ! Boo is a weak equivalence if and only if gnm : Anm ! Bnm
is a homotopy equivalence for all n; m 0. In addition, he observes
the surprising fact that this definition is equivalent to requiring that
Ao;m+1 ! Bo;m+1 xMmvBoo MvmAoo be a fibration for all n -1.
As before, Aoo is called fibrant if Aoo ! *= is a fibration where *= is
the bicosimplicial space which is a constant point.
Note that toth and totv preserve matching spaces, that is, M m( toth Aoo) =
tot (MvmAoo) and M n( totv Aoo) = tot (MhnAoo). Furthermore, since
tot preserves fibre products, it follows from 4.1 that:
Lemma 5.4. If g : Aoo ! Boo is a fibration, then toth g, totv g, and
tot g are all fibrations. In particular, if Aoo is fibrant, then toth Aoo,
totv Aoo, and tot Aoo are all fibrant.
This lemma and 4.2 imply:
Proposition 5.5. If g : Aoo ! Boo is a map of bicosimplicial spaces
with Aoo and Boo fibrant and if either toth g or totvg is a weak equiv-
alence, then tot g is a homotopy equivalence.
Let Xo ! Aoo be an augmented bicosimplicial space via a cosimplicial
map Xo ! A0o (or Xo ! Ao0). The above proposition specializes to:
Corollary 5.6. If Xo ! Aoo is an augmented bicosimplicial space
with Xo and Aoo fibrant and if Xo ! tothAoo (or Xo ! totvAoo) is a
weak equivalence, then tot Xo ! tot Aoo is a homotopy equivalence.
The cosimplicial examples of section 4 have analogues for bicosim-
plicial spaces. In particular, pullbacks of fibrations are fibrations, if B
is a fibrant space then the constant bicosimplicial space B___is fibrant,
isomorphisms are fibrations, and if Boo is fibrant then the projection
Aoo x Boo ! Aoo is a fibration.
Definition 5.7. A bicosimplicial space Aoo is grouplike if each fixed
row Aom and each fixed column Ano is a grouplike cosimplicial space.
Proposition 5.8. A surjective homomorophism Aoo ! Boo of grou-
16
plike bicosimplicial spaces is a fibration. In particular, a grouplike bi-
cosimplicial space is fibrant.
Proof: As in Bousfield-Kan , page 276, An+1 o ! Bn+1 o xMnhBoo
MhnAoo is a surjective homomorphism of grouplike cosimplicial spaces,
therefore, a fibration. ||
17
6. Contractions.
Let Xo be a cosimplicial object augmented by d0 : X-1 ! X0 . The
augmented cosimplicial object X-1 ! Xo is said to admit a left con-
traction if for n -1 there are maps s-1 : Xn+1 ! Xo such that the
usual cosimplicial identities are satisfied, that is:
didj = djdi-1 ifi > j
sidj = djsi-1 ifi > j; = 1 ifi = j or j - 1; = dj-1 si-1 ifi < j
sk sj = sj-1 si ifi < j :
Similarly, it said to admit a right contraction if for n -1 there are
maps sn+1 : Xn+1 ! Xn such that the usual cosimplicial identities are
satisfied.
Let d : X-1 ! tot Xo be the map induced by the augmentation.
A_left (or right) contraction gives a map __s: Xo ! X__-1 defined by
sn = s-1 . . .s-1 : Xn ! X-1 (or __sn= s0 . .s.n). If s = tot __*
*s:
tot Xo ! X-1 , then it is easy to see that sd : X-1 ! tot Xo ! X-1
is the identity. We shall show that ds : tot Xo ! X-1 ! tot Xo is
homotopic to the identity. Thus, d : X-1 ! tot Xo is a homotopy
equivalence whenever X-1 ! Xo admits a contraction.
The lemma below is left as an exercise.
Lemma 6.1. If W is a space and Xo and Y o are cosimplicial spaces,
then map (W; hom (Xo ; Y o)) = hom (W x Xo ; Y o) where map denotes
the simplicial mapping space.
Hence, a homotopy between ds and 1 is a map
H : hom (o; Xo ) ! hom (1 x o; Xo ) :
This homotopy is defined as follows. Let ss : 1 x n-1 ! n be the
map defined by: ss(0; i) = 0, ss(1; i) = i+1, and ss((a0; : :;:an ); (b0; : :;:*
*bn )) =
(ss(a0; b0); : :;:ss(an ; bn )). Similarly, let oe : 1 x n-1 ! n be de-
fined by oe(0; i) = i, oe(1; i) = n, and oe((a0; : :;:an ); (b0; : :;:*
*bn )) =
(oe(a0; b0); : :;:oe(an ; bn )). If X-1 ! Xo has a left contraction, define
H on an m-simplex f : m x o ! Xo by (Hf )n = s-1 f n+1 (1 x ss) :
m x 1 x n ! m x n+1 ! Xn+1 ! Xn . If it has a right contrac-
tion, define H by (Hf )n = sn+1 f n+1 (1 x oe).
Check that ss(1 x dj-1 ) = djss, ss(1 x sj-1 ) = sjss, oe(1 x dj) = djoe,
and oe(1 x sj) = sjoe.
18
Then it is easy to see that Hf is in hom (1 x o; Xo )n . It is also
apparent that H is a simplicial map. In the case of a left contraction,
H0 = ds and H1 = 1, and in the case of a right contraction, H0 = 1 and
H1 = ds.
The standard examples of contractions are as follows. Let T be a
triple and X ! T oX the canonical resolution.
1) A left contraction for T X ! T (T oX) is defined for n -1 by
s-1 = (T n+1 x) : T (T n+2 X) ! T (T n+1 X).
2) A right contraction for T X ! T o(T X) is defined for n -1 by
sn+1 = T n+1 (X) : T n+2 (T X) ! T n+1 (T X).
3) A constant object X ! X__admits both left and right contractions
given by s-1 = 1 and sn+1 = 1.
4) Let E be a space over B with E and B fibrant. The Rector
complex E ! (B x -)o(E) of example 4 in section 2 admits a
left contraction. For n -1, define it by letting s-1 : (B x
-)n+2 (E) ! (B x -)n+1 (E) be projection on the last n + 2
factors. Hence, E ! (B x -)1 (E) is a map over B which is
a homotopy equivalence where the target is a fibration. Notice
that the maps s-1 are not maps over B and hence the homotopy
inverse (B x -)1 (E) ! E is not a map over B.
Definition 6.2. Let T : C___! C___be a triple with structure maps j : 1 !
T and : T 2! T . A right T -module is a covariant functor M : C___! B___
together with a natural transformation ' : M T ! M such that the
following commute.
M T -'! M M T T M'-! M T
M j " % 1 # M # '
M M T -'! M
If M is a right T -module and S : B___! A__is a covariant functor, then
SM is also a right T -module via S'.
Of course, one can define a left T -module in the obvious way and one
has:
Proposition 6.3. If M is a right (left) T -module and X ! T oX is the
canonical resolution, then M X ! M T oX (M X ! T oM X) admits a
left (right) contraction.
Proof: Define s-1 : M T n+1 X ! M T nX by s-1 = '(T nX) and sn :
T n+1 M X ! T nM X by sn = T n'(x). ||
19
Notice that examples 1 and 2 above are special cases of 6.3 As a less
trivial example, we use 6.3 to show that two constructions of Bousfield-
Kan are equivalent.
__
Let_R X be the set of formal sums r[x] where rfflR, xfflX, and r = 1.
Then R X is contained in RX_ and it is clear that the triple_structure
of R restricts_to one for R . Furthermore, R is a right R module via
' : RR ! R defined by '(r[s[x]])_= rs[x]. We claim that there is
a homotopy equivalence R 1 X ! R1 X.
__o __o __m+1
Consider the bicosimplicial_space_Ro(R_ X) with_Ro(R_ X)nm = Rn+1 (R X*
*).
It has augmentations_RoX ! Ro(R_oX)_and R oX ! Ro(R oX). Fur-
thermore, RoX; R_oX, and Ro(R oX) are all grouplike, hence fibran-
t. (To make R oX _grouplike,_ choose a basepoint in X and convert
the_ affine spaces R m+1 X__into groups.) Since each Rn+1 is a right
R -module, RoX__! totvRo(R oX) is a weak equivalence and hence
R1 X ! tot Ro(R oX) is a homotopy equivalence. On the other hand,
by Bousfield-Kan , Y ! R1 (Y_)_is a homotopy_equivalence if Y is a
simplicial_R-module.__Hence, R oX ! R1 (R oX) is a weak equivalence
and R 1 X ! tot Ro(R oX) is a homotopy equivalence.
20
7. The homotopy pullback property.
In this section, we prove:
Proposition 7.1. Let E be a space over B with E and B fibrant. Then
BR1 (E) is a homotopy pullback of
R1 E
#
B ! R1 B :
The proof of 7.1 consists of two stages. First, we show that, in the
case of a product bundle B x F ! B, there is a homotopy equivalence
B x R1 (R) ! BR1 (B x F ). Second, we apply the fact that BR1 is
the fibrewise localization on product bundles to establish 7.1.
The natural map BR(B xF ) = B xR(B xF ) ! B xRF is compatible
with the triple structures of BR and R. Hence, it defines a map :
BRo(B x F ) ! B x Ro(F ) and a map 1 : BR1 (B x F ) ! B x R1 (F )
over B.
Lemma 7.2. If B is fibrant, then 1 : BR1 (B x F ) ! B x R1 (F ) is
a homotopy equivalence.
Proof: It follows from example 6 in section 4 that 1 is a map of
fibrations over B. In order to show that 1 is a homotopy equivalence,
it is sufficient to show that for every choice of a basepoint in B the
resulting map of fibres is a homotopy equivalence. (This requires that B
be fibrant.) Inspection shows that the fibre of BRo(B x F ) ! B__is the
canonical resolution R(Bx-)o(F ) associated to the triple F 7! R(BxF )
with evident structure maps F ! R(B x F ) and R(B x R(B x F )) !
R(B xF ). Furthermore, induces the map of fibres ff : R(B x-)o(F ) !
Ro(F ) coming from the map of triples R(B x F ) ! RF . Hence, we need
to show that ff1 : R(B x -)1 (F ) ! R1 (F ) is a homotopy equivalence.
Consider the bicosimplicial spaces Aoo = R(Bx-)o(RoF ) and Ro(RoF )
with Anm = R(B x -)n+1 (Rm+1 F ) and Ro(RoF )nm = Rn+1 (Rm+1 F ).
There are augmentations fi = R(B x -)o(j) : R(B x -)o(F ) ! Aoo,
fl = j : RoF ! Aoo, ffl = Ro(j) : RoF ! Ro(RoF ), and ffi = j : RoF !
21
Ro(RoF ) which fit into a commutative diagram.
F - ! R(B x -)o(F ) -ff! RoF
# # fi # ffl
RoF - fl! Aoo -! Ro(RoF ) :
ffi
That ff1 is a homotopy equivalence follows from the fact that the
induced maps ffl1 , ffi1 , fl1 and fi1 are homotopy equivalences. Since
RoF , R(B x-)o(F ), Aoo and Ro(RoF ) are all grouplike, they are fibrant
and we can apply Corollary 5.6.
Example 1 in section 6 shows that RoF ! totv Ro(RoF ) is a weak
equivalence and hence ffl1 : R1 F ! tot Ro(RoF ) is a homotopy e-
quivalence. Example 2 shows that RoF ! tot hRo(RoF ) is a weak
equivalence and hence ffi1 : R1 F ! tot Ro(RoF ) is a homotopy equiv-
alence.
To show that Rm+1 R ! R(B x -)o(Rm+1 F ) admits a contraction, it
suffices to treat the case m = 0. But ' = (F )R(ss2) : R(B x RF ) !
R(RF ) ! RF makes R into a left R(B x -)- module. Hence, there is
a right contraction and fl1 is a homotopy equivalence.
The left contraction in example 1 of section 6 is a linear contraction
of R ! R(RoF ). Since R(B x RoF ) = RB R(RoF ), we get a left
contraction of R(B x -)(F ) ! R(B x -)(RoF ) and, by induction, a
left contraction of R(B x -)n+1 (F ) ! R(B x -)n+1 (RoF ) for all n 0.
Hence, fi1 is a homotopy equivalence. ||
The proof of 7.2 being complete, we recast 1.2 in a more convenient for-
m. Define a map : BxRo(F ) ! BRo(BxF ) by (b; r0[r1[. . .[rn [f ]] . .].]) =
(b; r0[b; r1[b; . .[.rn [b; f ]] . .].]). Since the composition : B x
Ro(F ) ! BRo(B x F ) ! B x Ro(F ) is the identity, we get:
Lemma 7.3. If B is fibrant, then 1 : B x R1 (F ) ! BR1 (B x F ) is
a homotopy equivalence.
Consider what a homotopy pullback of
E
#
X - ! B
22
is. The Rector complex gives a homotopy equivalence E ! (Bx-)1 (E)
and the target is a fibration whenever E and B are fibrant. Let
(B x -)1 (E)|X - ! (B x -)1 (E)
# #
X - ! B
be a pullback diagram. Then a homotopy pullback is any fibration Y
over X which is homotopy equivalent to (B x -)1 (E)|X via maps over
B.
Form the bicosimplicial space Boo = BRo(B x -)o(E) with Bnm =
BRn+1 (B x-)m+1 (E). If we ignore coface operators, Boo is the product
of B___and a grouplike object. Hence, Boo is fibrant.
Form the bicosimplicial space Coo with Cnm = B x Rn+1 ((B x
-)m (E)) with horizontal operators coming from Rn+1 and vertical oper-
ators coming from embedding Cno into Rn+1 ((B x -)o(E)). If we forget
coface operators, then Coo is the product of B___and a grouplike object,
hence, Coo is fibrant.
The map in 7.3 gives a map : Coo ! Boo over B and a weak equiv-
alence toth : toth Coo ! toth Boo where toth Com = B x R1 ((B x
-)m (E)). Hence, tot : tot Coo ! tot Boo is a homotopy equiva-
lence.
By Bousfield-Kan , the map R1 (X x Y ) ! R1 X x R1 Y is always a
homotopy equivalence. Hence, there is a weak equivalence toth Coo !
Eo over B where Em = B x (R1 B x -)m (R1 E). There is a pullback
diagram
Eo ! (R1 B x -)o(R1 E)
# #
B__ ! R1_B___
Therefore, Eo is fibrant, tot Eo is the pullback (R1 B x -)1 (R1 E)|B ,
and there is a homotopy equivalence
tot Coo ! (R1 E x -)1 (R1 B)|B :
Since E ! (Bx-)o(E) admits a contraction, so does each BRn+1 (E) !
BRn+1 (B x -)o(E), BRo(E) ! tot vBoo is a weak equivalence, and
BR1 (E) ! tot Boo is a homotopy equivalence.
That BR1 (E) is a homotopy pullback of
R1 E
#
B - ! R1 B
23
is a consequence of these homotopy equivalences which are maps over
B, BR1 (E) ! tot Boo tot Coo ! (R1 B x -)1 (R1 E)|B .
24
References
(1) A.K. Bousfield, On the homotopy spectral sequence of a cosim-
plicial space, Amer. Jour. of Math., 109 (1987), 361-394.
(2) A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions
and Localizations, Springer-Verlag, 1972.
(3) W.G. Dwyer, to appear.
(4) J. Lannes, Sur la cohomogie modulo p des p-groupes Abeliens
elementaires, in Homotopy Theory, Proc. Durham Symp. 1985,
edited by E. Rees and J.D.S. Jones, Cambridge U. Press, 1987.
(5) H.R. Miller, The Sullivan conjecture on maps from classifying
spaces, Ann. of Math., 120 (1984), 39-87.
William Dwyer Haynes Miller
University of Notre Dame Mass. Inst. of Technology
Notre Dame, Indiana Cambridge, Massachusetts
Joseph Neisendorfer
University of Rochester
Rochester, New York
25