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On Newstead's Mayer-Vietoris argument in characteristic 2

Christopher Scaduto, Matthew Stoffregen

Abstract

Consider the moduli space of framed flat $U(2)$ connections with fixed odd determinant over a surface. Newstead combined some fundamental facts about this moduli space with the Mayer-Vietoris sequence to compute its betti numbers over any field not of characteristic two. We adapt his method in characteristic two to produce conjectural recursive formulae for the mod two betti numbers of the framed moduli space which we partially verify. We also discuss the interplay with the mod two cohomology ring structure of the unframed moduli space.

On Newstead's Mayer-Vietoris argument in characteristic 2

Abstract

Consider the moduli space of framed flat connections with fixed odd determinant over a surface. Newstead combined some fundamental facts about this moduli space with the Mayer-Vietoris sequence to compute its betti numbers over any field not of characteristic two. We adapt his method in characteristic two to produce conjectural recursive formulae for the mod two betti numbers of the framed moduli space which we partially verify. We also discuss the interplay with the mod two cohomology ring structure of the unframed moduli space.

Paper Structure

This paper contains 5 sections, 8 theorems, 26 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Getting started with the genus 1 decomposition
  4. Applying the Mayer-Vietoris argument
  5. Computations for the genus 2 decomposition

Key Result

Theorem 1

Write $h_r^g = \dim H^r(N_g^\#;\mathbb{Z}/2)$. Then we have the following: in which $m_r^g$ is the coefficient of $t^r$ in the polynomial $(1+t^3)^{2g}$. Further:

Figures (8)

  • Figure 1: Comparison of the $\mathbb{Z}/2$ and $\mathbb{Q}$ betti numbers of the framed moduli space. The $\mathbb{Z}/p$ betti numbers for $p$ prime, $p\neq 2$ are the same as the $\mathbb{Q}$ betti numbers. In each column half the betti numbers are listed; the rest are obtained by Poincaré duality. For example, the $\mathbb{Z}/2$ betti numbers of $N_2^\#$ are 1,0,1,5,5,5,5,1,0,1.
  • Figure 2: Genus 1 data. The notation $a_b^c$ stands for a linear map $\mathbb{F}^b\longrightarrow \mathbb{F}^c$ of rank $a$. All entries are computed from the first column from relations in Section \ref{['sec:preliminaries']}, except for $\nu_2^1$ and $\nu_3^1$ (boxed) -- see Lemma \ref{['lemma:nu1']}.
  • Figure 3: The $E_2$-page in the Leray-Serre spectral sequence for $N_2^\#$
  • Figure 4: The maps $\lambda_2^{1,1}$, $\lambda_3^{1,1}$ and $\lambda_4^{1,1}$. Each vector space has been replaced by a dot $\bullet$. The dimension of each vector space is written as a superscript of each $\bullet$. The notation $\oplus 2$ in the lower left pane indicates that the map $\lambda_3^{1,1}$ consists of two copies of the depicted map. The lone lower dot in the upper left pane of $\lambda_2^{1,1}$ comes from the domain of $\iota_1^1\otimes\nu_1^1$.
  • Figure 5: In the left hand pane, we have simply redrawn the above expansion of $\lambda_r^{1,g}$ with a dot $\bullet$ replacing the name of each vector space. The computation of all the left hand (red) maps in this pane allows us to replace $\lambda_r^{1,g}$ with the map $\psi_r^{1,g}$ defined in the right hand pane.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Lemma 3
  • Lemma 4
  • proof : Proof of (i)-(ii) in Thm. \ref{['theorem:main']}
  • Lemma 5
  • proof