Topological comparison theorems for Bredon motivic cohomology

Abstract

We prove equivariant versions of the Beilinson-Lichtenbaum conjecture for Bredon motivic cohomology of smooth complex and real varieties with an action of the group of order two. This identifies equivariant motivic and topological invariants in a large range of degrees.

Motivic analogues of MO and MSO

“…The following results about topological realization functors can be found in [HK11,HVØ16]. The application discussed herein is an observation of the current author.…”

“…As an aside, I would like to point out that the authors of [HKO11] use the greek letter γ instead of σ. The reason for this difference is an aesthetic one, although σ is also used in [HVØ16] in place of γ. However, in [HVØ16] the authors use a Voevodsky type grading.…”

“…The reason for this difference is an aesthetic one, although σ is also used in [HVØ16] in place of γ. However, in [HVØ16] the authors use a Voevodsky type grading. I prefer the grading convention of [HKO11].…”

Bredon motivic cohomology of the complex numbers

“…Bredon motivic cohomology (introduced in [8] and [9]) is a generalization of motivic cohomology to the setting of smooth varieties with finite group action. Part of a larger group of motivic C 2 -invariants, Hermitian K-theory, and motivic real cobordism play an essential role in equivariant motivic homotopy theory.…”

Motivic homotopical Galois extensions

“…This was followed by the study of equvariant motivic spectra by Hu, Kriz, and Ormsby in [24], who used them to study Karoubi's Hermitian K-theory and the motivic cobordism spectrum as C 2 -spectra. Recently, there has been a serious push to develop solid foundations for equivariant motivic homotopy theory, in particular, by Heller, Krishna, Ormsby, Voineagu, Østvaer and others (see for example [5], [12], [13], [14], [23]).…”