The group of homotopy self equivalences of a union of \((n-1)\)-connected \(2n\)-manifolds
In an earlier paper [\textit{W. D. Barcus} and \textit{M. G. Barratt}, Trans. Am. Math. Soc. 88, 57-74 (1958; Zbl 0095.16801)], methods were developed for studying classifications of extensions of a certain type of fixed map. This yielded information on the groups \({\mathcal E}(X)\) of pointed, homotopy-self-equivalences of \(X=K \cup_\alpha e^{q+1}\). When \(X\) is an \((n-1)\)-connected \(2n\)-manifold, or a single \(e^{2n}\) attached to a wedge of \(S^n\), calculations were made by \textit{P. J. Kahn} [Math. Ann. 180, 26-47 (1969; Zbl 0177.26101)]. The present note extends this admirably to the case of a wedge of 2 spaces of the type treated by P. J. Kahn. For such complexes there is a quadratic form \(Q(\alpha)\), which is related to the cup product form [\textit{C. T. C. Wall}, Ann. Math., II. Ser. 75, 163-189 (1962; Zbl 0218.57022)]. If \(X\) and \(Y\) do not have the same homotopy-type and the forms are non-singular, the map \([X\vee Y,X\vee Y] \to[X \vee Y,X\times Y]\) induces a faithful representation of \({\mathcal E}(X\vee Y)\) onto a quotient of an iterated semi-direct product made from homotopy groups of the \(n\)-skeletons and the groups \({\mathcal E}(X)\) and \({\mathcal E}(Y)\). The authors also treat the case where \(X\equiv Y\). These results are carefully illustrated with examples. For the quaternionic projective plane, \({\mathcal E}(HP^2\vee HP^2)\) is a dihedral extension of \(Z/4 \oplus Z/4\). For the Cayley projective plane, they show that \({\mathcal E} (CP^2\vee CP^2)\) is a semi-direct product of \((Z/2)^{10}\) and \(Z/2\).
scientific article; zbMATH DE number 1286737
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