Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bredon motivic cohomology of the real numbers

Bill Deng, Mircea Voineagu

TL;DR

This work computes the $C_2$-equivariant Borel motivic cohomology ring and the Bredon motivic cohomology groups of the real numbers with $ frac12$ coefficients, showing that the Bredon cohomology ring of $ r$ forms a proper subring of the $RO(C_2\times\Sigma_2)$-graded cohomology of a point. The authors organize the computations as modules over the $RO(C_2)$-graded cohomology of a point and establish a detailed ring structure: a positive cone generated by $x_i,y_i$ and a negative cone generated by $ heta$, together with an invertible periodicity element $\,\kappa_2$. They compute the Bredon cohomology of $rac{E}{C_2}$ and of the real numbers via the $C_2\times\Sigma_2$ topological isotropy sequence and Betti realization, showing that the realization maps are isomorphisms or injections in many regions of the bigrading. The results extend to real closed fields, with cycle maps linking motivic and topological invariants, and provide a comprehensive description of how the motivic cohomology of $ r$ sits inside the richer equivariant setting, highlighting the role of $rac{E}{C_2}$ in encoding Borel and genuine Bredon data.

Abstract

Over the real numbers with $\Z/2-$coefficients, we compute the $C_2$-equivariant Borel motivic cohomology ring, the Bredon motivic cohomology groups and prove that the Bredon motivic cohomology ring of the real numbers is a proper subring in the $RO(C_2\times C_2)$-graded Bredon cohomology ring of a point. This generalizes Voevodsky's computation of the motivic cohomology ring of the real numbers to the $C_2$-equivariant setting. These computations are extended afterwards to any real closed field.

Bredon motivic cohomology of the real numbers

TL;DR

This work computes the -equivariant Borel motivic cohomology ring and the Bredon motivic cohomology groups of the real numbers with coefficients, showing that the Bredon cohomology ring of forms a proper subring of the -graded cohomology of a point. The authors organize the computations as modules over the -graded cohomology of a point and establish a detailed ring structure: a positive cone generated by and a negative cone generated by , together with an invertible periodicity element . They compute the Bredon cohomology of and of the real numbers via the topological isotropy sequence and Betti realization, showing that the realization maps are isomorphisms or injections in many regions of the bigrading. The results extend to real closed fields, with cycle maps linking motivic and topological invariants, and provide a comprehensive description of how the motivic cohomology of sits inside the richer equivariant setting, highlighting the role of in encoding Borel and genuine Bredon data.

Abstract

Over the real numbers with coefficients, we compute the -equivariant Borel motivic cohomology ring, the Bredon motivic cohomology groups and prove that the Bredon motivic cohomology ring of the real numbers is a proper subring in the -graded Bredon cohomology ring of a point. This generalizes Voevodsky's computation of the motivic cohomology ring of the real numbers to the -equivariant setting. These computations are extended afterwards to any real closed field.
Paper Structure (17 sections, 41 theorems, 207 equations, 4 figures)

This paper contains 17 sections, 41 theorems, 207 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Equivariant motivic homotopy theory
  3. Bredon motivic cohomology
  4. $RO(C_2)$-graded Bredon cohomology of a point
  5. Our Results
  6. $RO(C_2\times \Sigma _2)$-graded Bredon cohomology of a point
  7. $RO(C_2\times \Sigma _2)$-graded Bredon cohomology groups of a point
  8. $RO(C_2\times \Sigma _2)$-graded Bredon cohomology ring of a point
  9. Positive Cone; i.e. $p,b,q\geq 0$.
  10. $C_2\times\Sigma_2$ topological isotropy sequence
  11. $\Sigma_2$-equivariant topological spaces
  12. The case $b+q<0.$
  13. The case $b\geq 0$, $b+q\geq 0$.
  14. The case $b\geq 0$ and $b+q<0$ revisited
  15. Bredon motivic cohomology of $\mathbf{E} C_2$
  16. ...and 2 more sections

Key Result

Proposition 1

$H^{a,b}(\mathds{R},\mathds{Z}/2)\simeq H^{a-b+b\epsilon} _{Br}(pt,\mathds{Z}/2)$ for any $a\in \mathds{Z}, b\geq 0$. Moreover, as rings, where the last expression is the positive cone of the Bredon cohomology of a point (i.e. $a+p\geq 0$, $p\geq 0$ in Figure TEST) with $deg(x_2)=(1,1)$ and $deg(y_2)=(0,1)$. The realization maps are monomorphisms, but not isomorphisms if $b+1<a\leq 0$.

Figures (4)

  • Figure 2: Regions of $H^{*+*\sigma}_{Br}(\mathds{R},\mathds{Z}/2)$ determined by (parts) of the $RO(C_2)-$ graded Bredon cohomology of $EC_2$ and $\tilde{E}C_2$. The degrees of the displayed elements are $|\alpha|=-1+\sigma$, $|\theta|=2-2\sigma$.
  • Figure 3: Regions of $H^{\star, b+q\sigma}_{C _2}(\mathds{R},\mathds{Z}/2)$ determined by $\mathbf{E} C_2$, Betti realization into the $RO(C_2\times \Sigma_2)$-graded Bredon cohomology of a point, and $\widetilde{\mathbf{E}} C_2$. The degree of the displayed element is $|\kappa_2|=(-2+2\sigma, -1+\sigma)$.
  • Figure 4:
  • Figure 5:

Theorems & Definitions (66)

  • Definition 1: HOV1
  • Proposition 1
  • Proposition 2
  • proof
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 3
  • Proposition 4
  • ...and 56 more