The equivariant stable homotopy theory around isometric linear maps

TL;DR

This paper develops a new equivariant homotopy theory for spaces of isometries between representations, extending classical non-equivariant results and exploring their topological and algebraic implications.

Contribution

It constructs a novel tower of G-spectra for equivariant isometries, linking to Miller's non-equivariant splitting and proposing conjectures on deeper connections and K-theory.

Findings

Constructed a tower of G-spectra reflecting non-equivariant splitting

Derived explicit topological and geometric descriptions of the tower

Proposed conjectures on interactions with Miller's splitting and K-theory

Abstract

The non-equivariant topology of Stiefel manifolds has been studied extensively, culminating in a result of Miller demonstrating that a Stiefel manifold splits stably to a wedge of Thom spaces over Grassmannians. Equivariantly, one can consider spaces of isometries between representations as an analogue to Stiefel manifolds. This concept, however, yields a different theory to the non-equivariant case. We first construct a variation on the theory of the functional calculus before studying the homotopy-theoretic properties of this theory. This allows us to construct the main result; a natural tower of G-spectra running down from equivariant isometries which manifests the pieces of the non-equivariant splitting in the form of the homotopy cofibres of the tower. Furthermore, we detail extra topological properties and special cases of this theory, developing explicit expressions covering the tower's geometric and topological structure. We conclude with two detailed conjectures which provide an avenue for future study. Firstly we explore how our theory interacts with the splitting of Miller, proving partial results linking in our work with Miller's and conjecturing even deeper connections. Finally, we begin to calculate the equivariant K-theory of the tower, conjecturing and providing evidence towards the idea that the rich topological structure will be mirrored on the K-theory level by a similarly deep algebraic structure.