The group of homotopy self-equivalences of some compactCW-complexes

Abstract

For a space X, there is the problem of determining its group of homotopy self-equivalences ¢(X), i.e. those (based) homotopy classes of maps from X to itself which are invertible with respect to composition. One approach has been to look at the Postnikov decomposition of X: starting with the observation that for an Eilenberg-Mac Lane space 8(K(n, n))_~ Autrc, one can study g for induced fibrations and derive certain properties of 8(X). But there seems little hope to actually determine 8(X) in this way for spaces which have infinitely many non-zero homotopy groups, such as almost all compact spaces. In fact, the only results known to the author in this context are the homotopy-type classification of lensspaces (Franz, J.H.C. Whitehead, Olum) and the calculation of 8 for the pseudo-projective planes (Olum [3]). We describe here an approach which seems more suitable in the compact case. To any cellular map from one C W complex to another belongs its chain-homomorphism, i.e. the homomorphism induced on the (cellular) chaincomplexes. For chain-homomorphisms as well as for maps, there is the notion of homotopy, and a function is defined from homotopy classes to chain-homotopy classes. One can ask when this function is a one-to-one correspondence. Since our interest lies with non-simply connected spaces, it is better to look instead at the homomorphisms induced on the chaincomplexes of the universal coverings. We study a class of spaces for which the so-obtained function is a bijection; the class includes all compact complexes X such that nk(X)=0 for 1 <k < dimX. For such spaces g(X) and the group ~(X) of simple selfequivalences have purely algebraic descriptions. The paper starts with stating the basic classification theorem. Its proof is an application of the obstruction theory for non-simply connected spaces and is deferred to the end. It specializes to two quite explicit descriptions of the normal subgroup g~(X) of g(X) consisting of those self-equivalences which induce the identity on nl. We also discuss briefly