Stable self-homotopy equivalences

Abstract

In this paper we develop a new method for the study of structural properties of the group stable self-homotopy equivalences. Using this new approach we give more efficient proofs for some classical theorems, together with a series of new results.

On smooth manifolds with the homotopy type of a homology sphere

“…Due to the naturality of suspension isomorphism and Bott periodicity, we can identify Aut(K(X)) with Aut(K(Σ 8i X)). Hence, the natural map from Aut(X) to Aut(K(X)) factors through the group of stable self equivalences of X, which is equal to colim i Aut(Σ 8i X) (see for example [37], [38] and [39] for more details about the group of stable self equivalences). If X is a Z/q-homology sphere for a prime q, then all Betti numbers of X are less than or equal to 1.…”

The ring of stable homotopy classes of self-maps ofAn2-polyhedra

“…Such ring is usually denoted by End(X). Consequently, the group of stable self-homotopy equivalences of X, denoted E S (X), is just the group of units of End(X), and both of these algebraic structures associated to spectra have been studied by several authors, [15,14,18,19].…”

The ring of stable homotopy classes of self-maps of $A_n^2$-polyhedra

“…Both End(X) and E S (X) have been studied by several authors, [11,12,15,16]. In particular, it is known that for finite CW-complexes, End(X) and the endomorphism ring of H * (X) regarded as a graded module are isomorphic module torsion, [11,Corollary 3.2].…”