The algebraic transfer for the real projective space

Abstract

A chain-level representation of the Singer transfer for any left A-module is constructed. We prove that the image of the Singer transfer Tr⁎RP∞ for the infinite real projective space is a module over the image of the transfer Tr⁎ for the sphere. Further, the algebraic Kahn–Priddy homomorphism is an epimorphism from ImTr⁎RP∞ onto ImTr⁎ in positive stems. The indecomposable elements hˆi for i≥1 and cˆi, dˆi, eˆi, fˆi, pˆi for i≥0 are detected, whereas the ones gˆi for i≥1 and Dˆ3(i), pˆi′ for i≥0 are not detected by the Singer transfer Tr⁎RP∞. This transfer is shown to be not monomorphic in every positive homological degree. The transfer behavior is also investigated near “critical elements”. We prove that Kameko's squaring operation on the domain of Tr⁎RP∞ is eventually isomorphic. This phenomenon leads to the so-called “stability” of the Singer transfer for the infinite real projective space under the iterated squaring operation.