On equivariant self-homotopy equivalences of \(G\)-CW complexes (Q5931521)

For any map \(f : A \to B\), with cofibre \(C_f\), there is a corresponding Puppe sequence of the form \(\cdots \to [\Sigma A, C_f] \to [C_f, C_f] \to [B, C_f] \to [A, C_f]\). This gives rise to the so-called ``Barcus-Barratt'' method for studying the group of self-homotopy equivalences of \(C_f\), denoted \({\mathcal E}(C_f) \subseteq [C_f, C_f]\). The method consists of attempts to refine this Puppe sequence so as to display \({\mathcal E}(C_f)\) as a term in some (exact) sequence of groups. This approach was used in [\textit{W. D. Barcus} and \textit{M. G. Barratt}, Trans. Am. Math. Soc. 88, 57-74 (1958; Zbl 0095.16801)], to study the case in which \(A\) is a sphere. A number of authors have used the same approach to study various other cases. For example in [\textit{S. Oka, N. Sawashita}, and \textit{N. Sugawara}, Hiroshima Math. J. 4, 9-28 (1974; Zbl 0284.55013)] the authors study the case in which \(A\) is a suspension and in addition further hypotheses hold.NEWLINENEWLINENEWLINEIn the current paper, a similar approach is taken. The main difference is that the equivariant case is studied here. Let \(G/H^+\) denote the quotient of a finite group \(G\) by some subgroup \(H\), with a disjoint basepoint added. Let \(A = G/H^+\wedge S^n\), the \(n\)-fold reduced suspension of \(G/H^+\). Let \(f : A \to B\) be a \(G\)-map, and let \({\mathcal E}_G(C_f)\) denote the group of \(G\)-equivariant self-homotopy equivalences of \(C_f\). If \(n \geq 2\) and \(B\) is a \(G\)-\(1\)-connected suspension of dimension \(\leq n-1\), then \({\mathcal E}_G(C_f)\) is displayed as the central term of a short exact sequence, as in the Barcus-Barratt result (Theorem 3.5). The authors also obtain several concrete computational results from this. They compute the following: \({\mathcal E}_G(G/H^+\wedge S^n)\) for finite abelian \(G\); \({\mathcal E}_{{\mathbb Z}_2}(C_f)\) for certain maps \(f : {\mathbb Z}_2^+\wedge S^{n+k} \to {\mathbb Z}_2^+\wedge S^n\); \({\mathcal E}_{{\mathbb Z}_6}(C_f)\) for certain maps \(f : {\mathbb Z}_6^+\wedge S^{n+k} \to {\mathbb Z}_2^+\wedge S^n\).