A generalization of the stable EHP spectral sequence

TL;DR

This paper generalizes the stable EHP spectral sequence to arbitrary vector bundles, establishing new convergence results and limits, and extends classical theorems to broader contexts with spectral sequence techniques.

Contribution

It introduces a generalized spectral sequence for vector bundles, extending Mahowald and Lin's work, and analyzes limits and convergence in new settings.

Findings

Established strong convergence in certain cases

Generalized Lin's theorem for compact base spaces

Determined limits using Adams spectral sequence for BO

Abstract

For any vector bundle, we define an inverse system of spectra. In the case of a trivial bundle over a point, the homotopy groups of the filtration quotients give rise to the stable EHP spectral sequence, as was shown by Mahowald. The limit was determined by Lin. We also obtain a spectral sequence and show strong convergence in certain special cases. For compact base spaces, we obtain a generalization of Lin's theorem. In the case of the universal bundle over BO, we can also determine the limit by means of an Adams spectral sequence. This turns out to be quite different from the compact case. We also obtain partial results for the universal bundle over BSO.