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$RO(C_2\times C_2)$-graded cohomology ring of a point and applications

Bill Deng, Mircea Voineagu

TL;DR

This work computes the $RO(C_2 imes ext{Σ}_2)$-graded cohomology ring of a point, decomposing it into a positive cone, two mixed cones (Type I and II), and a negative cone, with explicit generators, relations, and nilpotent elements. It establishes the ring structure and periodicities via isotropy cofibrations, and uses these results to determine the $RO(C_2 imes ext{Σ}_2)$-graded cohomology of $E_{ ext{Σ}_2}C_2$, including a precise top-level description with invertible periodicity elements. The paper then applies the theory to the Bredon motivic cohomology of real numbers, showing a decomposition into a motivic-like base plus a nilpotent cone NC, and it realises NC within the full BR-cohomology of a point. The appendix provides detailed multiplicative computations and Poincaré-series counts essential for the structural results. Overall, the results illuminate the interplay between motivic and equivariant phenomena in the $C_2 imes ext{Σ}_2$ setting and extend Voevodsky’s positive-cone identification to a Klein four-group context, with concrete consequences for real motivic cohomology and related spaces.

Abstract

We describe the main properties of the $RO(C_2\times Σ_2)$-graded cohomology ring of a point and apply the results to compute the subring of motivic classes given by the Bredon motivic cohomology of real numbers and to compute $RO(C_2\times Σ_2)$-graded cohomology ring of $E_{Σ_2}C_2$. This generalizes Voevodsky's identification of motivic cohomology of real numbers with the positive cone of $RO(C_2)$ graded cohomology of a point.

$RO(C_2\times C_2)$-graded cohomology ring of a point and applications

TL;DR

This work computes the -graded cohomology ring of a point, decomposing it into a positive cone, two mixed cones (Type I and II), and a negative cone, with explicit generators, relations, and nilpotent elements. It establishes the ring structure and periodicities via isotropy cofibrations, and uses these results to determine the -graded cohomology of , including a precise top-level description with invertible periodicity elements. The paper then applies the theory to the Bredon motivic cohomology of real numbers, showing a decomposition into a motivic-like base plus a nilpotent cone NC, and it realises NC within the full BR-cohomology of a point. The appendix provides detailed multiplicative computations and Poincaré-series counts essential for the structural results. Overall, the results illuminate the interplay between motivic and equivariant phenomena in the setting and extend Voevodsky’s positive-cone identification to a Klein four-group context, with concrete consequences for real motivic cohomology and related spaces.

Abstract

We describe the main properties of the -graded cohomology ring of a point and apply the results to compute the subring of motivic classes given by the Bredon motivic cohomology of real numbers and to compute -graded cohomology ring of . This generalizes Voevodsky's identification of motivic cohomology of real numbers with the positive cone of graded cohomology of a point.

Paper Structure

This paper contains 18 sections, 54 theorems, 361 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $C_2\times \Sigma_2$-topological isotropy sequence
  3. The $C_2\times\Sigma_2$-topological isotropy sequence
  4. $\Sigma_2$-equivariant classifying spaces
  5. $RO(C_2\times \Sigma_2)$- graded cohomology of a point
  6. $RO(C_2\times \Sigma_2)$- graded cohomology ring of a point
  7. The ring structure of the middle level of $\ul{H}_{Br,K}^\star(pt)$
  8. Mixed Cone of Type I
  9. The Mixed Cone of Type II
  10. The Negative Cone
  11. $RO(C_2\times \Sigma_2)$-graded cohomology of $E_{\Sigma_2}C_2$
  12. The Middle and Lower Levels of $\ul{H}_{Br,K}^\star({E_{\Sigma_2}{C_2}};\ul{\mathbb{Z}}_2)$
  13. The ring structure of the middle level of $\ul{H}_{Br,K}^\star(E_{\Sigma_2}C_2)$
  14. The Top Level of $\ul{H}_{Br,K}^\star({E_{\Sigma_2}{C_2}};\ul{\mathbb{Z}}_2)$
  15. Bredon motivic cohomology ring of real numbers
  16. ...and 3 more sections

Key Result

Theorem 1.0.1

We have a $\mathds{M}^{C_2}_2$-algebra isomorphism The description of the cohomology classes is given in Section 3. The component $nilpotents_1$ has only zero products among its elements, contains only nilpotents and it is completely described in Section 5. This is a subring in with $nilpotents_1\subset nilpotents_2$ a strict inclusion and the isomorphism in the display being $\mathds{M}^{C_2}_2

Figures (1)

  • Figure 1: Since $L=\left| \{(i,j) : j-i =-a'-B+p+q\} \cap \{ (i,j): i \in [0,B-2], j\in[0,q-1] \}\right|$, we can compute $L$ by counting the number of points in the intersection of the lattice $[0,B-2]\times[0,q-1]$ with the line $j= i-a'-B+p+q.$ Here, we show the case where $a'=1,p=q=5,$ and $B=7.$ We see that the intersection consists of $3$ points, which is precisely equal to $B-p+a',$ corresponding to the case in the top line of the previous display. As $a'$ varies, the line $j= i-a'-B+p+q$ shifts up and down, since all the other variables are fixed. Then we see that the cardinality of the intersection coincides precisely with the corresponding cases in the previous display.

Theorems & Definitions (84)

  • Theorem 1.0.1
  • Theorem 1.0.2
  • Theorem 2.2.1: Kr,HK1
  • Theorem 2.2.2
  • proof
  • Theorem 2.2.3
  • Theorem 2.2.4: Vanishing
  • proof
  • Proposition 3.1.1
  • Proposition 3.1.2
  • ...and 74 more