Self maps of \(\Sigma ^ k{\mathbb{C}}P^ 3\) for k\(\geq 1\)
If X is a pointed space, then \(M(X)=[X,X]\), the set of (based) homotopy classes of self-maps of X, becomes a monoid under multiplication. If, in addition, X is a co-H-space, then M(X) is a ring. Particularly interesting is the group E(X) of units in M(X). In the paper under review, the group E(X) resp. the ring M(X) are explicitly calculated for the suspension \(\Sigma^ k{\mathbb{C}}P^ 3\) of the complex projective 3-space, \(k\geq 1\). The descriptions of E(X) and M(X) are in terms of generators and relations, the formulae are too complicated to be presented here. As a corollary of his results, the author is able to describe precisely those endomorphisms of \(H_*(\Sigma^ k{\mathbb{C}}P^ 3)\) which are realized by a continuous self-map. In the proof the author calculates the homotopy groups \(\pi_ q(\Sigma^ k{\mathbb{C}}P^ 3)\) in a certain range of dimensions.
scientific article; zbMATH DE number 4205396
scientific article