Stable maps to projective spaces
Let \(HP^{\infty}\) (resp. \(Q^{\infty})\) be the infinite dimensional quaternionic projective space (resp. the infinite dimensional quaternionic quasi-projective space). In this paper the author studies the natural homomorphism from \(\{\) X,Y\(\}\) to \(Hom(H_*(X)\), \(H_*(Y))\) when Y is \(HP^{\infty}\) or \(Q^{\infty}\), all homology groups have integer coefficients and where \(\{\) X,Y\(\}\) means the based homotopy set of stable maps from X to Y. In a previous paper [Publ. Res. Inst. Math. Sci. 20, 971-976 (1984; Zbl 0556.55007)] the author determined the image of the induced homomorphism from \(\{\Sigma^{4n}HP^{\infty}\), \(HP^{\infty}\}\) to \(Hom(H_*(HP^{\infty})\), \(H_*(HP^{\infty}))\). In that paper, for \(Q^{\infty}\) there was a mistake. The author says that this mistake was pointed out by \textit{H. Oshima} who proved a corrected theorem (unpublished). In this paper the author gives a slightly different (as the author declares) statement of this theorem. Then, using a similar method, the author constructs various stable maps between projective spaces so that the homology homomorphisms induced by those stable maps give the basis of the image from stable maps to their induced homology homomorphisms. The method used (that introduced in the op. cit. of the author) can be extended to cases where the source spaces of stable maps satisfy certain conditions.
infinite dimensional quaternionic projective space
infinite dimensional quaternionic quasi-projective space
scientific article; zbMATH DE number 4073963
scientific article