K-theory and formality

TL;DR

This paper computes equivariant K-theory for formal isotropy actions, introduces a new characterization of equivariant formality, and relates weak formality to the surjectivity of the forgetful map, broadening understanding of equivariant topology.

Contribution

It provides a spectral sequence comparison approach to equivariant K-theory, introduces a new characterization of equivariant formality, and links weak formality to the surjectivity of the forgetful map.

Findings

Constructs a spectral sequence map from Hodgkin's Künneth spectral sequence to Borel cohomology.

Establishes equivalence between weak equivariant formality and surjectivity of the forgetful map.

Main structure theorem parallels Borel equivariant cohomology, with accessible proof and new corollaries.

Abstract

We compute the equivariant K-theory with integer coefficients of an equivariantly formal isotropy action, subject to natural hypotheses which cover the three major classes of known examples. The proof proceeds by constructing a map of spectral sequences from Hodgkin's K\"unneth spectral sequence in equivariant K-theory to that in Borel cohomology. A new characterization of equivariant formality appears as a consequence of this construction, and we are now able to show that weak equivariant formality in the sense of Harada--Landweber is equivalent with integer coefficients to surjectivity of the forgetful map under a standard hypothesis. The main structure theorem is formally similar to that for Borel equivariant cohomology, which appears in the author's dissertation/dormant book project and whose proof is finally made accessible in an appendix. The most generally applicable corollary of the main theorem for rational coefficients depends on a strengthening of the characterization of equivariant formality due to Shiga and Takahashi, which appears as a second appendix.