Twisted self-homotopy equivalences

Abstract

This paper studies the group G(A X B) of (homotopy classes of) self-homotopy equivalences of a product A X B of two connected CW homotopy associative ϋf-spaces A and B. It establishes the existence of an exact sequence of multiplicative groups 1-> [A A B, A X B]-> G{A X B)-> GL(2, A TJ)-» 1 provided that io[A x B, A X B]oqo[A A B, A X B] = 0, where <?: A x 2?->A Λ B is the cofibration induced by the inclusion i: A\/ B-> Ax B of the sum into the product. The entry GL(2, ΛJJ) is the group of invertible matrices rh Λ-fh A A h AB \ KPΊJ)-7 7 \JlBA hβB/ with entries h ΣJ in the homotopy sets Λu = [I, J] for I, J = A, B, where matrix multiplication is defined by (huXku) = (h IA ok A i + hiB°kBj) in terms of composition o and the operation + in the homotopy sets [/,/], and where the multiplicative unit is

Self-homotopy equivalences of product spaces

“…Proof: It is su cient to modify slightly the original Sieradski's proof -see 15], and to note that the image consists exactly of matrices described in the proposition.…”

Certain numbers on the groups of self-homotopy equivalences

“…Then, E(X) is a group with the operation given by the composition of homotopy classes. E(X) has been studied extensively by various authors, including Arkowitz [1], Maruyama [4], Lee [12], Oka [13], Rutter [15], Sawashita [16], and Sieradski [17]. Several subgroups of E(X) have also been studied.…”

Reducibility of self-homotopy equivalences

“…This leaves RP 7 ×RP 7 as the only open case among non-simply connected rank 2 H-spaces (cf. also [6,15,16] and [9, problem 5]). …”

“…as a product or wedge of simpler spaces, and describe self-equivalences of X by means of maps between its factors. There have been several attempts to achieve this goal, most notably by Booth and Heath [6] for general products and by Sieradski [16] for products of H-spaces. We will not go into details of their results but, generally speaking, both approaches, and indeed all the others, require two types of assumption.…”