The group of self-homotopy equivalences - a survey

On the group of homotopy equivalences of a manifold

“…For a differential manifold M the group E(M ) is comparable with the group Π 0 Diff(M ) of isotopy classes of diffeomorphisms of M . In fact, via the J -homomorphism there is a striking similarity between these groups as is shown in § 10 below.Still there is little known on the group E(M ) in the literature; only very specific examples are computed, see [2]. This paper contains on the one hand general results on the structure of the group E(M ), see §1, .…”

“…the group of automorphisms of the homotopy type of X, can be regarded as the homotopy symmetry group of the space X. In the literature there has been a lot of interest in the computation of such groups; compare for example the excellent survey article of M. Arkowitz [2].…”

“…Still there is little known on the group E(M ) in the literature; only very specific examples are computed, see [2]. This paper contains on the one hand general results on the structure of the group E(M ), see §1, .…”

ON THE GROUP AutΩ(X)

“…When endowed with the operation induced by the composition of maps, Aut(X) becomes a group, called the group of self-homotopy equivalences of X. This group has been extensively studied (see the survey [1] or the recently published monograph [11]). …”

A bound for the nilpotency of a group of self homotopy equivalences

“…Given a pointed space X of the homotopy type of a CW-complex, let E(X) denote the group of based homotopy classes of self homotopy equivalences of X ( [1] is an excellent survey on this object). A considerable amount of work has been dedicated to obtaining finiteness properties, not only of E(X), but also of certain interesting subgroups which preserve additional geometrical structure (see for example [2], [5], [6], [8]).…”