Equivariant self-homotopy equivalences of product spaces
Let \(G\) be a finite group. Recall that a based \(G\)-space is \(G\)-space whose base-point is \(G\)-fixed. For pointed \(G\)-spaces \(X\) and \(Y\), let \(\mathrm{map}^G(X,Y)\) denote the space of all based \(G\)-maps from \(X\) to \(Y\), and we denote by \([X,Y]_G\) the space of all based \(G\)-homotopy classes of based \(G\)-maps from \(X\) to \(Y\). Let \(\mathrm{aut}^G(X)\) be the space of all based \(G\)-equivariant self-homotopy equivalences of \(X\), and let \(\mathrm{Aut}^G(X)\) denote the set of based \(G\)-equivariant homotopy classes of \(\mathrm{aut}^G(X)\). Note that this is a group whose multiplication is induced from the composition of maps. For a \(G\)-self-homotopy equivalence \(f:X\times Y\to X\times Y\), we denote by \(f_{XX}\) and \(f_{YY}\) the maps given by \[ f_{ZZ}=p_Z\circ f\circ \iota_Z :Z\to Z \quad \text{for }Z=X,Y, \] where \(p_Z:X\times Y \to Z\) and \(\iota_Z:Z\to X\times Y\) denote the projection and the inclusion for \(Z=X\) or \(Y\). Then a based \(G\)-self-homotopy equivalence \(f\in \mathrm{Aut}^G(X\times Y)\) is called \textit{reducible} if \(f_{XX}\in \mathrm{aut}^G(X\times Y)\) and \(f_{YY}\in \mathrm{aut}^G(X\times Y)\). For \(Z=X\) or \(Y\), let \(\mathrm{Aut}^G_{Z}(X\times Y)\) denote the subgroup of \(\mathrm{Aut}^G(X\times Y)\) given by \[ \mathrm{Aut}^G_{Z}(X\times Y)=\{f\in \mathrm{Aut}^G(X\times Y): p_Z\circ f=p_Z\}. \] Let \(\mathrm{aut}(X)\) be the space of all based self-homotopy equivalences of \(X\) and let \(\mathrm{aut}_1(X)\) denote the path-component of the identity map \(\mathrm{id}_X:X\to X\) of the space \(\mathrm{aut}(X)\). In this paper, the authors study the group \(\mathrm{Aut}^G(X\times Y)\), and they prove that \[ \mathrm{Aut}^G(X\times Y)=\mathrm{Aut}^G_X(X\times Y)\cdot \mathrm{Aut}^G_Y(X\times Y) \] if all \(g\)-self-homotopy equivalences of \(X\times Y\) are \(G\)-reducible. Moreover, under the same assumption as above, they also show that there is a split short exact sequence \[ 1\to [X,\mathrm{aut}_1(Y)]_G\to\mathrm{Aut}^G(X\times Y) \stackrel{\phi}{\longrightarrow} \mathrm{Aut}^G(X)\times \mathrm{Aut}^G(Y)\to 1 \] if \([Y,\mathrm{aut}_1(X)]_G=\{*\}\), where \(\phi (f)=(f_{XX},f_{YY})\) for \(f\in \mathrm{Aut}^G(X)\). \par Note that the above two results have already been obtained by Heath and Booth when \(G\) is a trivial group [\textit{P. R. Heath}, Quaest. Math. 19, No. 3--4, 433--451 (1996; Zbl 0872.55005) and \textit{P. I. Booth} and \textit{P. R. Heath}, Lect. Notes Math. 1425, 17--31 (1990; Zbl 0707.55009)]. So the two results may be regarded as one possible generalization of the result due to Heath-Booth. The authors also generalize the above results for the products \(X_1\times X_2\times \cdots \times X_n\) when each \(X_i\) is a based \(G\)-space.
scientific article; zbMATH DE number 7646061