A -algebra SPECTRAL SEQUENCE FOR FUNCTION SPACES
W. G. Dwyer, D. M. Kan, J. H. Smith and C. R. Stover
University of Notre Dame
Massachusetts Institute of Technology
Purdue University
University of Chicago
x1. Introduction
1.1 The main result.
Given a map f : K ! L of pointed CW complexes, let hom f(K; L) denote the
pointed space of pointed maps K ! L, with f as the base point. Recall that a -
algebra is a ( 1)-graded group with an action of the primary homotopy operations
(for example, for any pointed topological space M there is a homotopy -algebra
ss*M = {ssiM}1i=1). Given a map t : X ! Y of -algebras, let hom t(X; Y )
denote the "function -algebra" defined below in 3.4. Then (3.5) there is a natu*
*ral
map b : ss* hom f(K; L) ! hom ss*f(ss*K; ss*L) of -algebras, which (3.6) is an
isomorphism whenever K has the homotopy type of a wedge of spheres of dimensions
1. Our main result is a generalization of this fact. Let hom (p)-(-; Y ) (p *
* 0)
denotes the p'th right derived functor, in the sense of Quillen [3], of the abo*
*ve
functor hom -(-; Y ) from " -algebras over Y " to " -algebras".
1.2 Theorem. Let f : K ! L be a map of pointed connected CW complexes.
Then
(1) there exists a natural second quadrant spectral sequence {Ep;qr} which *
*is
closely related (in the sense of [1, IX, x5]) to ss* hom f(K; L). The E*
*2-term
of this spectral sequence is given by
E0;q2= hom (0)ss*f(ss*K; ss*L)q = hom ss*f(ss*K; ss*L)q q 1
Ep;q2= hom (p)ss*f(ss*K; ss*L)q q p 1
and the edge homomorphism
ss* hom f(K; L) ! E0;*1! E0;*2= hom ss*f(ss*K; ss*L)
coincides with the above mentioned map b.
(2) In view of 4.3, this spectral sequence converges strongly to ss* hom f(*
*K; L)
if L has only a finite number of non-trivial homotopy groups or if the
-algebra ss*K has finite cohomological dimension in the sense of 4.1.
______________
This research was in part supported by the National Science Foundation.
Typeset by AM S-T*
*EX
1
2 W. G. DWYER, D. M. KAN, J. H. SMITH AND C. R. STOVER
Examples of pointed connected CW complexes with a homotopy -algebra of
finite cohomological dimension are ([2] and 4.4) wedges of spheres (of dimensio*
*ns
1) and finite dimensional CW complexes wwith the homotopy type of a K(ss; 1).
From these one can construct other such spaces using the following lemma, which
is an immediate consequence of [2, 7.5] and 4.4.
1.3 Lemma. If * ! W ! X ! Y ! * is a short exact sequence of -algebras
and W and Y have finite cohomological dimension, then so does X.
1.4 Organization of the paper. We first (in x2) prove 1.2(1) only for the c*
*ase
in which the map f : K ! L is trivial (i.e., maps all of K to the base point of*
* L).
In x3 we indicate what changes have to be made in order to remove this restrict*
*ion.
In the last section (x4) we observe that the abelian group hom (p)t(X; Y )q *
*can
be interpreted as the p'th Quillen cohomology of X with local coefficients in t*
*he
"q-fold loops on Y ". This immediately implies 1.2(2).
x2. Proof of 1.2(1) (special case)
In this section we prove 1.2(1) for the case that the map f : K ! L is trivi*
*al,
i.e., f maps all of K to the base point * 2 L.
We start with a brief review of the notion of a -algebra, which involves a
"category of homotopy operations" which is slightly different from, although
equivalent to, the one of [4].
2.1 The category of homotopy operations. This will be the category
which has as objects the pointed CW complexes with the homotopy type of a finite
wedge of spheres of dimensions 1 and which has as maps the homotopy classes
of (pointed) maps between them. Note that
(1) the category is pointed and has finite coproducts (i.e. finite wedges)*
* but
not products, and
(2) the category comes with a smash functor i : x ! which sends
an object (U; V ) 2 x to the object
U ^ V = (U x V )=((U x *) _ (* x V )) 2
and which preserves coproducts in each variable, i.e., the functors (U *
*^ -) :
! and (- ^ V ) : ! send coproducts to coproducts.
Using the category we now define
2.2 -algebras. Let Sets *denote the category of pointed sets. A -algebra
then can be defined as a contravariant functor ! Sets* which sends coproducts
to products, and a map of -algebras as a natural transformation between two
such functors. The resulting category of -algebras will be denoted -al .
This definition implies that, for every object X 2 -al
(1) X* = *, where * denotes the point in both categories and Sets *, and
(2) the values of X on the objects of are, up to isomorphism, determined *
*by
the values of X on the spheres Sn , (n 1). These values will be denoted
by Xn.
A -algebra SPECTRAL SEQUENCE FOR FUNCTION SPACES 3
In view of Hilton's analysis of the homotopy groups of wedges of spheres [5, XI*
*] one
can thus consider a -algebra X as a ( 1)-graded group {Xn}1n=1, with Xn abelian
for n > 1, together with Whitehead product homomorphisms [-; -] : Xp Xq !
Xp+q-1 (p; q 1) and composition functors (-).ff : Xp ! Xr (ff 2 ssrSp; 1 < p <*
* r)
which satisfy all the identities that hold for the Whitehead product and compos*
*ition
operations on the higher homotopy groups of pointed topological spaces, and a l*
*eft
action of X1 on the Xn (n > 1) which commutes with these operations.
An obvious example of a -algebra is thus provided by
2.3 The homotopy -algebra of a pointed topological space. Given a
pointed topological space M, the functor ! Sets* which sends an object U 2
to the set of homotopy classes of (pointed) maps U ! M is easily seen to be a
-algebra. Since (2.2) this -algebra is completely determined by the homotopy
groups ssnM (n 1) and the action of the (primary) homotopy operations on them,
we often denote this -algebra by ss*M.
Next we define
2.4 Abelian -algebras. A -algebra X will be called abelian if there exists
a "multiplication map" X x X ! X 2 -al which turns X into an abelian group
object in -al . For every integer n 1, the restriction (X xX)n = Xn xXn ! Xn
then is the multiplication map for Xn and hence (2.2) the original multiplicati*
*on
map on X, if it exists at all, is unique. A straightforward calculation now yi*
*elds
that a -algebra X is abelian iff
(1) X1 is abelian and acts trivially on the Xn, (n 1),
(2) all Whitehead products in X are trivial, and
(3) all composition functions in X are homomorphisms.
An abelian -algebra will be called strongly abelian if all composition funct*
*ions
are trivial, i.e., if it is just a ( 1)-graded abelian group.
2.5 Example. Given objects Y 2 -al and U 2 , let Y U : ! Sets* be the
functor given by Y UV = Y (V ^ U) for all V 2 . Then it is not dificult to see*
* that
Y U is an abelian -algebra.
Using these abelian -algebras Y U we can now construct
2.6 Function -algebras. For two objects X, Y 2 -al , let hom *(X; Y ) :
! Sets* [4, A.3] be the functor which sends an object U 2 to the set of maps
X ! Y U 2 -al . Then again it is not difficult to verify that hom *(X; Y ) is*
* an
abelian -algebra.
If the variables in hom *(-; -) are homotopy -algebras of pointed CW complex*
*es
(2.3) then these function -algebras are closely related to
2.7 Homotopy -algebras of function spaces. Suppose that K and L are
pointed CW complexes. If hom *(K; L) denotes the pointed space of pointed maps
K ! L (with the trivial map as the basepoint) one can construct a natural map
h : ss* hom *(K; L) ! hom *(ss*K; ss*L) 2 -al
by sending (the homotopy class of) a map U ! hom *(K; L) to (the homotopy class
of) the corresponding map K ! hom *(U; L) and then composing the resulting
4 W. G. DWYER, D. M. KAN, J. H. SMITH AND C. R. STOVER
map ss*K ! ss* hom *(U; L) 2 -al with the isomorphism ss* hom *(U; L) ~=(ss*L*
*)U
which sends (the homotopy class of) a map V ! hom *(U; L) to (the homotopy
class of) its adjoint V ^ U ! L.
A straightforward calculation now yields
2.8 Proposition. The natural map (2.7) h : ss* hom (K; L) ! hom *(ss*K; ss*L) 2
-al is an isomorphism whenever K has the homotopy type of a (not necessarily
finite) wedge of spheres of dimensions 1.
Finally we are ready for a
2.9 Proof of 1.2(1) for f : K ! L the trivial map. Let VoK be the simplicial
resolution of K by wedges of spheres of dimensions 1 described in [4, x2]. The
desired second quadrant spectral sequence then will be the [1, X, x6] homotopy
spectral sequence {Ep;qr} of the cosimplicial pointed space hom *(VoK; L). If V*
*oK
denotes the realization of VoK [4, x3], then [4, x3] the canonical map VoK ! K
is a homotopy equivalence and [1, p. 335] Tot(hom *(VoK; L)) ~= hom *(VoK; L).
Consequently the spectral sequence is closely related [1, IX, x5] to ss* hom *(*
*K; L).
That Ep;q2= hom (p)*(ss*K; ss*L)q for q p 0 and q 1 follows readily from *
*2.8,
[3], [1, X, x7] and the fact [4, x2] that ss*VoK is a free (and hence cofibrant*
*) simplicial
-algebra and its projection ss*VoK ! ss*K onto ss*K is a weak equivalence of
simplicial -algebras.
Finally, a direct calculation shows that hom (0)*(ss*K; ss*L) = hom *(ss*K;*
* ss*L)
and that the edge homomorphism in the spectral sequence coincides with the map
b of 2.7.
x3. Proof of 1.2(1) (general case)
We now prove 1.2(1) without any restriction by generalizing the arguments of
x2. We start with reminding the reader of the existence of
3.1 A half smash functor in . The category (2.1) comes with a half
smash functor o : x ! which sends an object (U; V ) 2 x to the object
(U xV )=(*xV ) which also can be written as U ^V +, where V + denotes the point*
*ed
CW complex obtained from V by adding a disjoint basepoint. It clearly has the
following properties.
(1) Behavior in the first variable. For every object V 2 , the restrict*
*ion
functor (- o V ) : ! sends coproducts to coproducts.
(2) Behavior in the second variable. This is more complicated. For U; V 2 ,
there are natural maps
j p
U = U o * AE U o V = U ^ V +!- U ^ V
q
in such that qj = idand pj = *. There is also a map k : U ^V ! U oV 2
, which is not natural, such that pk = idand such that the resulting *
*map
j _ k : U _ (U ^ V ) ! U o V 2
is an isomorphism.
Next we generalize abelian -algebras (2.4) to
A -algebra SPECTRAL SEQUENCE FOR FUNCTION SPACES 5
3.2 -algebras with an action of a -algebra.. By an action of a -
algebra Y on an abelian -algebra A we mean a diagram in -al of the form
d
* ! A!-c B AE Y ! *
e
in which
(1) the right pointing arrows form an exact sequence, and
(2) de = id: Y ! Y .
One readily verifies that, under these conditions, the multiplication map of*
* A
(2.4) turns the map d : B ! Y into an abelian group object in the over category
-al =Y (which has as objects the maps X ! Y 2 -al and as maps the obvious
commutative triangles).
3.3 Example. Given objects Y 2 -al and U 2 , there is a natural action of
Y on the abelian (2.4) -algebra Y U given by the diagram
p* + j*
* ! Y U -! Y U AE Y ! *
q*
in which+p, q, and j are as in 3.1 and Y U+ denotes the -algebra such that
Y U (V ) = Y (V ^ U+ ) = Y (V o U) for all V 2 .
Using this natural action we now construct
3.4 Function -algebras. For a map t : X ! Y 2 -al let hom t(X; Y ) :
! Sets+* be the functor which sends an object U 2 to the set of the maps
X ! Y U 2 -al (3.3) whose composition with j* is t, pointed by the map q*t.
Then it is not difficult to verify that hom t(X; Y ) is an abelian -algebra. I*
*f t is
the trivial map, this construction clearly reduces to the one of 2.6.
As before (2.7) these function -algebras are related to
3.5 Homotopy -algebras of function spaces. If f : K ! L is a map of
pointed CW complexes and hom f(K; L) denotes the pointed space of pointed maps
K ! L, with the map f : K ! L as base point, one can construct a natural map
b : ss* hom f(K; L) ! hom ss*f(ss*K; ss*L) 2 -al
by sending (the homotopy class of) a map U ! hom f(K; L) to (the homotopy class
of) the corresponding map K ! hom *(U+ ; L) and then composing the resulting
map ss*K ! ss* hom *(U+ ; L) 2 -al with the isomorphism ss* hom *(U+ ; L) ~=
(ss*L)U+ which sends (the homotopy class of) a map V ! hom *(U+ ; L) to (the
homotopy class of) its adjoint V ^ U+ ! L.
Again a straightforward calculation yields
3.6 Proposition. The natural map (3.5) b : ss* hom f(K; L) ! hom ss*f(K; L) 2
-al is an isomorphism whenever K has the homotopy type of a (not necessarily
finite) wedge of spheres of dimensions 1.
And we conclude this section with a
6 W. G. DWYER, D. M. KAN, J. H. SMITH AND C. R. STOVER
3.7 Proof of 1.2(1) (general case). If, in the notation of 2.9, f0 : VoK ! L
denotes the composition of f with the projection VoK ! K, then the desired
spectral sequence is the homotopy spectral sequence of the cosimplicial pointed
space hom f0(VoK; L) (this is the same cosimplicial space as in 2.9, but with a
different basepoint). The rest of the proof now proceeds just as in 2.9, except*
* that
one has to use 3.6 instead of 2.8.
x4. Quillen cohomology of -algebras
In this last section we observe that the function -algebras of 3.4 are clos*
*ely
related to the
4.1 Quillen cohomology of -algebras. Given a map g : W ! X 2 -al
and an abelian -algebra A with X-action
d
* ! A ! B AE X ! *
let HQ (g; A) denote the associated abelian group of "derivations from W to A",
i.e., of maps h : W ! B 2 -al such that dh = g. For every integer p 0, the p*
*'th
Quillen cohomology group HpQ(X; A) of X with local coefficients in A then is the
abelian group obtained by applying to the identity map of X the p'th right deri*
*ved
functor (in the sense of Quillen [3]) of (3.2) the functor HQ (-; A) : -al =X !
Abelian groups . We will say that X has finite cohomological dimension if the*
*re
is an integer k 0 such that HqQ(X; A) = 0 for all q k and every abelian
-algebra A with an X-action.
As usual this definition of cohomology implies
4.2 Proposition. Let X 2 -al and let * ! A0! A ! A00! * be a short exact
sequence of abelian -algebras with an X-action. Then there is a natural long
exact sequence
0 ! H0Q(X; A0) ! . . .
! HpQ(X; A0) ! HpQ(X; A) ! HpQ(X; A00) ! Hp+1Q(X; A0) ! . . .
Another easy consequence of 4.1 is the following proposition, which immediat*
*ely
implies 1.2(2).
4.3 Proposition. Let t : X ! Y 2 -al . Then there is, for every p 0 and
q 1, a natural isomorphism hom (p)t(X; Y )q ~= HpQ(X; Y Sq), where the X-action
on Y Sq is the one induced by t from the natural Y -action on Y Sq (3.3).
We end with remarking that the notion of finite cohomological dimension of a
-algebra can be reduced to the same such notion for a simplicial ring. This fol*
*lows
readily from the next proposition (4.4) and the fact that [2, x8], at least in *
*positive
dimensions, the Quillen cohomology of a -algebra X with local coefficients in a
strongly (2.3) abelian -algebra is, apart from a shift in dimension, just ordin*
*ary
cohomology of the simplicial ring EFoX obtained by applying the enveloping ring
functor E [2, x3] to the standard free simplicial resolution [2, x2] FoX of X.
A -algebra SPECTRAL SEQUENCE FOR FUNCTION SPACES 7
4.4 Proposition. Let X 2 -al and k 0 be such that HqQ(X; A0) = 0 for all q
k and every strongly abelian -algebra A0 with an X-action. Then HqQ(X; A) = 0
for all q k + 1 and every abelian -algebra A with an X-action.
Proof. Let A be an abelian -algebra with an X-action and, for each s 0, let
A(s) A denote the sub -algebra whith an X-action such that A(s)n= 0 for n < s
and A(s)n= An for n s. As each quotient A(s)=A(s+1)(s 0) is a strongly abelian
-algebra with an X-action, 4.2 implies inductively that HqQ(X; A(s)) = 0 for all
s 0 and q k. Furthermore, A = lim A(s)and hence A fits into a short exact
sequence of abelian -algebras with an X-action
Y Y
* ! A ! A(s)! A(s)! * :
s s
The desired result now follows by applying 4.2 once again and noting that Quill*
*en
cohomology commutes with products.
References
[1] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizatio*
*ns, Lect. Notes
in Math. 304, Springer, Berlin, 1972.
[2] W. G. Dwyer and D. M. Kan, Homology and cohomology of -algebras, preprint (*
*Notre
Dame), 1989.
[3] D. G. Quillen, On the (co)-homology of commutative rings, Categorical Algeb*
*ra, Proc. Symp.
Pure Math. 17, Amer. Math. Soc., Providence, 1970, pp. 65-87.
[4] C. R. Stover, A van Kampen spectral sequence for higher homotopy groups, To*
*pology (to
appear).
[5] G. W. Whitehead, Elements of homotopy theory, Grad. Texts in Math. 61, Spri*
*nger, Berlin,
1978.
University of Notre Dame, Notre Dame, Indiana 46556
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Purdue University, West Lafayette, Indiana 47907
University of Chicago, Chicago, Illinois 60637
Processed August 27, 1992