The group of self-homotopy equivalences - a survey

Abstract

If X is an object in a category ¢, then let Eq(X) denote the set of morphisms f : X --~ X which are equivalences, i.e., there is a morphism g : X ~ X with f o g = I x and g o f = Ix. Composition of morphisms gives Eq(X) the structure of a group, called the group of self equivalences of X, with the identity morphism the unit of the group. For many categories the group of equivalences is a familiar and well-studied object. For example, if C is the category of groups, Eq(G) is Aut G, the automorphism group of G, if C is the category of topological spaces, Eq(X) is Homeo(X), the homeomorphism group of X, if C is the category of C°°-manifolds, Eq(M) is Diffeo(M), the diffeomorphism group of M, if C is the category of Riemannian manifolds, Eq(M) is Isom(M), the group of isometrics of M, and so on. The category frequently studied in algebraic topology is ffh ' the category whose objects are topological spaces with base point and whose morphisms are based homolopy classes of based maps. For this category the group of equivalences of X is denoted $(X) and called the group (of homotopy classes) of self-hornotopy equivalences of X. Thus $(X) is the analogue for the homotopy category of the automorphism group of a group, the homeomorphism group of a space, etc., and therefore can be regarded as a kind of homotopy symmetry group of a space. In this paper we give a survey of known results on $(X).