On the stable homotopy of the real projective space of even low dimension (Q1083716)

The purpose of this paper is to determine the group structure of the stable homotopy group \(\{P^{2n},P^{2n}\}\) for \(n\leq 4\), where \(P^ n\) denotes the real n-dimensional projective space. As a corollary, the stable group of self-homotopy equivalences of \(P^{2n}\) for \(n\leq 4\) is determined. The first motivation of this work is to ascertain Adams' theorem about the uniqueness of the existence of the Kahn-Priddy map. Our method is to use the composition methods developed by Toda. In particular, we use the knowledge of the following: The order of the identity class of \(P^{2n}\), the order of the Kahn-Priddy map and the ring structure of the stable homotopy ring of spheres in low dimension.