Spaces with finitely many non-trivial homotopy groups

Abstract

It is well known that the homotopy category of connected CW-complexes X whose homotopy groups n;(X) are trivial for i> 1 is equivalent to the category of groups. One of the objects of this paper is to prove a similar equivalence for the connected CW-complexes X whose homotopy groups are trivial for i > n + 1 (where n is a fixed non-negative integer). For n = 1 the notion of crossed module invented by J.H.C. Whitehead [13], replaces that of group and gives a satisfactory answer. We reformulate the notion of crossed module so that it can be generalized to any n. This generalization is called an ‘n-cat-group’, which is a group together with 2n endomorphisms satisfying some nice conditions (see 1.2 for a precise definition). With this definition we prove that the homotopy category of connected CW-complexes X such that Xi(X) = 0 for i > n + 1 is equivalent to a certain category of fractions (i.e. a localization) of the category of n-cat-groups. The main application concerns a group-theoretic interpretation of some cohomology groups. It is well known [lo, p. 1121 that the cohomology group H2(G; A) of the group G with coefficients in the G-module A is in one-to-one correspondence with the set of extensions of G by A inducing the prescribed G-module structure on A. Use of n-cat-groups gives a similar group-theoretic interpretation for the higher cohomology groups H”(G; A) and H”(K(C, k); A) where K(C, k) is an Eilenberg-MacLane space with k 2 1. In [8] we proved that crossed modules could be used to interpret a relative cohomology group. Here we show that the notion of n-cat-group is particularly suitable to interpret some ‘hyper-relative’ cohomology groups. The usefulness of this last result appears in its application to algebraic Ktheory where it leads to explicit computations [4]. This was in fact our primary motivation for a generalization of crossed modules. Section 1 contains the definitions of n-cat-groups and of n-cubes of fibrations. There are two functors: 5 : (n-cubes of fibrations) + (n-cat-groups)