On a complex topological orientation for circle-equivariant K-theory

TL;DR

This paper establishes a complex topological orientation for circle-equivariant K-theory, linking projective space line indices with Fourier expansions, advancing the understanding of equivariant topological invariants.

Contribution

It introduces a novel complex topological orientation for Atiyah-Segal circle-equivariant K-theory, connecting geometric and Fourier-analytic structures.

Findings

Existence of a complex topological orientation for circle-equivariant K-theory.

Indexing of projective space lines via Fourier expansion.

Bridging geometric structures with harmonic analysis in equivariant topology.

Abstract

The principal result of this note is the existence of a complex topological orientation for Atiyah-Segal $\mathbb{T}$-equivariant K-theory which indexes the projective space of lines in complex (n+1)-space by the Fourier expansion $1 + q + \dots + q^n$.