Mod 2 cohomology of 2-configuration space of a closed surface and Stiefel--Whitney class

TL;DR

This work determines the mod $2$ cohomology of the ordered $2$-configuration space $C_{2}(M)$ for closed surfaces $M$ as $\Sigma_{2}$-representations and uses this to deduce the $H^{*}(\mathbb{R}P^{\infty};\mathbb{F}_{2})$-module structure of the unordered space $B_{2}(M)$. The approach combines Leray--Hirsch, the Fadell--Neuwirth fibration, and Serre spectral sequences arising from a Borel construction to relate the cohomology to a base $\mathbb{R}P^{\infty}$ and a diagonal class in $H^{*}(M\times M;\mathbb{F}_{2})$. The authors provide explicit decompositions for orientable and nonorientable surfaces, including detailed $\,\mathbb{F}_{2}[\Sigma_{2}]$-summands and the $\mathbb{F}_{2}[\alpha]$-module structures with $|\alpha|=1$, and conclude that the Stiefel--Whitney height equals $2$ for orientable $M$ and $3$ for nonorientable $M$. These results have implications for generalized van Kampen–Flores-type theorems via the Stiefel--Whitney height and enrich the understanding of configuration space cohomology in low dimensions.

Abstract

In this paper, we compute the singular cohomology groups $H^*(C_2(M);\mathbb{F}_2)$ of the ordered 2-configuration space $C_2(M)$ as $Σ_2$-representations. Using the result, we determine the mod 2 cohomology of the unordered 2-configuration space $B_2(M)$ as a $H^*(\mathbb{R}P^{\infty};\mathbb{F}_2)$-module. As a corollary of our computation, we see that the Stiefel--Whitney height of $M$ is $2$ or $3$ when $M$ is orientable or not, respectively.

Figures (8)

• Figure 1: $E^{\prime}_{2}$-term
• Figure 2: $E_2$-term
• Figure 3: $E_3$-term
• Figure 4: $E_\infty$-term
• Figure 5: $E_2$-term

Theorems & Definitions (7)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.1
• Lemma 2.1
• Proposition 2.1