The group of stable self-equivalences

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.…”

Realization problems for the group of homotopy classes of self-equivalences

“…It also has the practical use of determining how much the choice of k-invariants overdetermines the homotopytype of a space. G(X) is usually a complicated, non-abelian group, but we often have strong information -for example, we know in important cases, that it is finitely-presented (see [7]). …”

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The group of homotopy classes of homotopy equivalences from a space X to itself, which we write as G(X), has been studies in recent years by Arkowitz and Curjel [1,2], the author [6,7] and several others. It is a natural quotient of the associative H-space made from homotopy-equivalcnces from X to itself under composition, and it has the same position, in the homotopy category, as the group of automorphisms of a group in the category of groups.

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The stable group of self-homotopy equivalences of sphere bundles over the sphere

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The group of homotopy classes of homotopy equivalences of a CW-complex X with itself, denoted here as G(X), has been studied by various authors. Specially a stabilization of G{X) is defined by the suspension homomorphism G(X)-* G(ΣX), denoted here as G'(X)=lim N G(Σ N X), and has been studied by several authors [3,4,5,6]. Nevertheless, our examples for the computation of stable groups are not abundant.

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