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Mod 2 representation of the symmetric group of order 2 over cohomology groups of 2-configuration space of torus

Tomoki Tokuda

Abstract

In this article, we compute the mod 2 representarion of the symmetric group of order 2 over the singular cohomology groups of orderd 2-configuration space $C_{2}(T^{d})$ of the $d$-torus $T^{d}$ for $d\geq 1$. As applications of the computation, we determine the Stiefel-Whitney height of $C_{2}(T^{d})$ for any $d$, and determine $\mathbb{F}_{2}[Σ_{2}]$-module structure of the cohomology groups of the unordered 2-configuration space of the $d$-torus for $d=2,3$ using the Serre spectral sequence.

Mod 2 representation of the symmetric group of order 2 over cohomology groups of 2-configuration space of torus

Abstract

In this article, we compute the mod 2 representarion of the symmetric group of order 2 over the singular cohomology groups of orderd 2-configuration space of the -torus for . As applications of the computation, we determine the Stiefel-Whitney height of for any , and determine -module structure of the cohomology groups of the unordered 2-configuration space of the -torus for using the Serre spectral sequence.
Paper Structure (3 sections, 4 theorems, 13 equations, 7 figures)

This paper contains 3 sections, 4 theorems, 13 equations, 7 figures.

Table of Contents

  1. Introduction and Main Theorems
  2. Computation
  3. Applications

Key Result

Theorem 1.1

The representation of $\Sigma_{2}$ over $H^{i}({C_{2}({T^{d}})};{\mathbb{F}_2})$ is decomposed as following:

Figures (7)

  • Figure 1: Cell decomposition of $T^{3}$
  • Figure 2: $E_2$-term
  • Figure 3: $E_3$-term
  • Figure 4: $E_\infty$-term
  • Figure 5: $E_2$-term
  • ...and 2 more figures

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1