Certain subgroups of groups of self-pair homotopy equivalences

Abstract

Let epsilon(X) be the set of based homotopy classes of based self-homotopy equivalences of a CW-complex X. The concept of epsilon(X) is applied to the category of pairs and is extended to a general concept epsilon(alpha) for a map alpha : A -> B. In this study, epsilon(alpha) is generalized to epsilon(gamma)(alpha) for two objects alpha and gamma. Several generalized subgroups of epsilon(alpha) or epsilon(gamma)(alpha) are obtained and are combined to form an exact sequence. The exactness and the split property of this sequence is investigated. In particular, the sequence of a product space or a wedge space is demonstrated to be a split exact sequence. The split property and the exactness are used to completely compute those subgroups. (C) 2019 Elsevier B.V. All rights reserved.