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
PurdueUniversity
University of Chicago
x1. Introduction
1.1 The main result.
Givena map f : K ! L of pointed CW complexes, let hom f(K;L) denote the
pointed space of pointed maps K! L, with f asthe 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 = fssiM g1i=1). Given a map t : X ! Y of -algebras, let hom t(X; Y) denote
the "function -algebra" defined below in ?!?. Then (?!?) there is a natural
map b : ss hom f (K;L) ! hom ss f(ss K;ss L) of -algebras, which (?!?) is an
isomorphism whenever Khas 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 derivedfunctor, in the sense of Quillen [3], of the above
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 fEp;qrg whichis
closely related (in the sense of[1, IX, x5]) to ss hom f(K; L). The E2-*
*term
of this spectral sequence is given by
E0;q2= hom (0)ss(fss K;ss L)q =hom ss f(ss K;ss L)q q 1
Ep;q2= hom (p)ss(fss K;ss L)q q p 1
and the edge homomorphism
ss hom f(K; L) ! E0;1! E20;= hom ss f(ss K; ss L)
!! coincides with the above mentioned map b.
!! (2) In view of ?!?,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 ?!?.
!
This research was in part supported by the National Science Foundation.
Typeset by AMS-TEX