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An alternate computation of the stable homology of dihedral group Hurwitz spaces

Aaron Landesman, Ishan Levy

Abstract

We give an different proof of our result computing the stable homology of dihedral group Hurwitz spaces. This proof employs more elementary methods, instead of higher algebra.

An alternate computation of the stable homology of dihedral group Hurwitz spaces

Abstract

We give an different proof of our result computing the stable homology of dihedral group Hurwitz spaces. This proof employs more elementary methods, instead of higher algebra.

Paper Structure

This paper contains 15 sections, 21 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Outline of paper
  4. Acknowledgements
  5. Notation for Hurwitz spaces
  6. Cohomology of the two-sided $\mathcal{K}$ complex
  7. Review of the usual $\mathcal{K}$-complex
  8. Definition the two-sided $\mathcal{K}$-complex
  9. A nullhomotopy for two-sided $\mathcal{K}$ complexes
  10. A particular two-sided $\mathcal{K}$-complex
  11. Proof of \ref{['theorem:two-sided-g']}
  12. Computing the cohomology stabilized by a single monodromy
  13. Proof of \ref{['claim:vanishing']}
  14. Proof of \ref{['proposition:stabilize-by-one-element-trivial']}
  15. Computing the stable homology

Key Result

Theorem 1.0.1

Choose an odd prime $\ell$ and use $G$ to denote the dihedral group of order $2\ell$, $G : = \mathbb Z/\ell \mathbb Z \rtimes \mathbb Z/2 \mathbb Z$. Let $c \subset G$ denote the conjugacy class of order $2$ elements. There are constants $I$ and $J$ depending only on $G$ so that for $n > iI + J$ and

Figures (3)

  • Figure 1: The complex above depicts the summand $C_{\bullet,\bullet}$ (defined in \ref{['remark:z-summand']}) of the two sided $\mathcal{K}$-complex, $\mathcal{K}(k[g], A,k[g,g^{-1}])$.
  • Figure 2: The left is a picture of the Fox-Neuwirth/Fuks cell structure of an element of the first stable cohomology. The right hand side pictures some additional constraints we may impose on the cell structure, as explained in \ref{['example:h1-argument']}.
  • Figure 3: This is a picture of part of the two-sided $K$ complex used to compute the stable first cohomology, as relevant to \ref{['example:h1-argument']}.

Theorems & Definitions (52)

  • Theorem 1.0.1
  • Remark 1
  • Remark 2
  • Lemma 6
  • Remark 7
  • proof
  • Definition 8
  • Remark 9
  • Definition 10
  • Remark 11
  • ...and 42 more