The suspension order of the real even dimensional projective space
In the category of pointed topological spaces, denote the suspension of \(X\) by \(\Sigma X\), and the identity class of \(X\) by \(\iota_X\). The order of \(\iota_{\Sigma X}\in [\Sigma X,\Sigma X]\) is called the suspension order (or the characteristic) of \(X\). The order of \(\Sigma^{\infty}\iota_X\) is called the stable order of \(X\) [see \textit{H. Toda}, Ann. Math. 78, 300--325 (1963; Zbl 0146.18901)]. Let P\(^{n}\) be the real projective space of dimension \(n\). The main theorem in this paper states that the suspension order of P\(^6\) is \(8\). As an application of the theorem, the author concludes the truth of the conjecture in [\textit{J. Mukai}, J. Math. Soc. Japan 40, 53--63 (1988; Zbl 0637.55008)]: The suspension order of P\(^{2n}\) coincides with its stable order.
scientific article; zbMATH DE number 2124477