An equivariant generalization of the Miller splitting theorem

TL;DR

This paper proposes a conjectural equivariant extension of Miller's splitting theorem for Stiefel manifolds using a tower of G-spectra, with partial results and obstructions analyzed.

Contribution

It introduces a new tower of G-spectra aimed at generalizing Miller's splitting theorem in the equivariant setting, providing partial proofs and identifying obstructions.

Findings

Identifies a cohomological obstruction to splitting in most cases.

Proves the conjecture in certain special cases.

Develops a variation of the functional calculus with homotopy-theoretic properties.

Abstract

Let G be a compact Lie group. We build a tower of G-spectra over the suspension spectrum of the space of linear isometries from one G-representation to another. The stable cofibres of the maps running down the tower are certain interesting Thom spaces. We conjecture that this tower provides an equivariant extension of Miller's stable splitting of Stiefel manifolds. We provide a cohomological obstruction to the tower producing a splitting in most cases; however, this obstruction does not rule out a split tower in the case where the Miller splitting is possible. We claim that in this case we have a split tower which would then produce an equivariant version of the Miller splitting, we prove this claim in certain special cases though the general case remains a conjecture. To achieve these results we construct a variation of the functional calculus with useful homotopy-theoretic properties and explore the geometric links between certain equivariant Gysin maps and residue theory.