A -algebra SPECTRAL SEQUENCE FOR FUNCTION SPACES

1.1 The main result. Given a map f : K → L of pointed CW complexes, let homf (K,L) denote the pointed space of pointed maps K → L, with f as the base point. Recall that a Πalgebra 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 Π-algebra π∗M = {πiM}i=1). Given a map t : X → Y of Π-algebras, let homt(X,Y ) denote the “function Π-algebra” defined below in 3.4. Then (3.5) there is a natural map b : π∗ homf (K,L) → homπ∗f (π∗K,π∗L) of Π-algebras, which (3.6) is an isomorphism wheneverK has the homotopy type of a wedge of spheres of dimensions ≥ 1. Our main result is a generalization of this fact. Let hom – (–, Y ) (p ≥ 0) denotes the p’th right derived functor, in the sense of Quillen [3], of the above functor hom–(–, Y ) from “ Π-algebras over Y ” to “ Π-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,q r } which is closely related (in the sense of [1, IX, §5]) to π∗ homf (K,L). The E2-term of this spectral sequence is given by