On the group ${\mathcal E}\left[ X \right]$ of homotopy equivalence maps

Abstract

Let X be a CW-complex; we shall consider the group 2 s[x]formed by the homotopy classes of equivalence maps from X into itself with the operation induced by the composition of maps. It is clear to see that this group depends only on the homotopy type of X, hence should be determined by the known homotopy invariants of X. This is the problem which we shall try to study here. In fact, there exists a spectral sequence converging to S[X], whose initial terms are given, roughly speaking, by the cohomology of X and the automorphism group of its homotopy group.Besides the satisfaction of curiosity, the group S[X] seems to have other interests. For example, it operates canonically on the special cohomology group [l] of X

The group of self-equivalences of certain complexes

“…This leads to a collection of general results on the algebraic structure of the group of self-equivalences, as well as several explicit calculations, including the recovery of results due to Olum. This group has been studied by various authors, e.g. ArkowitzCurjel [1], Olum [7], Rutter [8], and Shih [9]-In general this group is non-abelian and quite often infinite.…”

“…The homomorphism Ψ is a very useful one in the general problem of determining Eq(X). In fact Eq{X) = ψ-\K) where K is the subgroup of Aut (Z m ), m = order of π λ (X) 9 consisting of the identity map and the inverse automorphism (x-> -x).…”

Self-Homotopy-Equivalences of a Space with Two Nonvanishing Homotopy Groups

“…For the simply connected case, the above result is shown by W. Shin [10] and later by Y. Nomura [8].…”

Homotopy and Topological Actions on Spaces with few Homotopy Groups

“…The discussion was an application of their results on homotopy classification. The first papers which dealt exclusively with the group of self-equivalences appeared in 1964 and were due to Kahn [11], Shih [18], and Arkowitz and Curjel [2,3]. The first general results relating the group ¿?…”

“…In §4, we show that if X is a stable 2-stage space such that the group R in Shih's exact sequence for l? (X) [18] is equal to 0, then any homotopy action of a group G on X is equivalent to a topological action. Finally, §5 examines the 2-stage approximation to Cooke's negative example.…”