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Farrell cohomology of the pure mapping class group of non-orientable surfaces

Nestor Colin

Abstract

For an odd prime $p$, we determine the $p$-primary component of the Farrell cohomology of the pure mapping class groups of a non orientable surface of genus $p$ with $k\geqslant 1$ marked points. To do this, we classify conjugacy classes of subgroups of order $p$ of the pure mapping class group of a non orientable surface of any genus with marked points. This is obtained by extending the notion of topological equivalence for surface kernel epimorphisms of non Euclidean crystallographic groups, adapting it to the setting of surfaces with marked points.

Farrell cohomology of the pure mapping class group of non-orientable surfaces

Abstract

For an odd prime , we determine the -primary component of the Farrell cohomology of the pure mapping class groups of a non orientable surface of genus with marked points. To do this, we classify conjugacy classes of subgroups of order of the pure mapping class group of a non orientable surface of any genus with marked points. This is obtained by extending the notion of topological equivalence for surface kernel epimorphisms of non Euclidean crystallographic groups, adapting it to the setting of surfaces with marked points.

Paper Structure

This paper contains 11 sections, 21 theorems, 67 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Subgroups of order p in N g k
  3. Conjugacy classes of subgroups of order p in Ngk
  4. Non-euclidean crystallographic groups.
  5. Surface kernel epimorphisms of NEC groups.
  6. Surface kernels and mapping class groups
  7. Surface kernels and t-tuples
  8. Normalizers of subgroups of order p in Ngk
  9. Birman-Hilden theory
  10. Case Npk, k=1,2
  11. The p-primary component of the Farrell cohomology of Ngk

Key Result

Theorem 1

Let $g\geqslant 3$ and $k\geqslant 1$. Then $\mathcal{N}_g^k$ contains a subgroup of order $p$ if and only if the Riemann-Hurwitz equation $g-2=p(h-2)+t(p-1)$ has an integer solution for $h$ and $t$ with $t\geq k, h\geq 1$.

Figures (2)

  • Figure 1: Loops generators of the fundamental group of $N_1^2$ and loops used in the punctured slides
  • Figure 2: Effects of $v_1$ on $\gamma_1$, $\gamma_2$, and $\delta$.

Theorems & Definitions (47)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • proof : Proof of Theorem \ref{['Thm:Main:Torsion']}
  • ...and 37 more