Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Big mapping class groups acting on homology

Federica Fanoni, Sebastian Hensel, Nicholas G. Vlamis

Abstract

We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.

Big mapping class groups acting on homology

Abstract

We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.

Paper Structure

This paper contains 21 sections, 37 theorems, 51 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Necessity of the conditions
  3. Structure of the paper
  4. Preliminaries
  5. Ends of a surface
  6. Homology of surfaces
  7. Homology classes of simple closed curves
  8. The Loch Ness Monster surface
  9. The infinite-degree integral symplectic group.
  10. The homology end filtration
  11. Flare surfaces and their homology
  12. The homology end filtration and its ultrafilters
  13. A homeomorphisms of the space of ends
  14. The homology end filtration and simple isotropics
  15. Characterizing the image of $\rho_S$ using simple isotropic elements
  16. ...and 6 more sections

Key Result

Theorem 1

Let $S$ be the Loch Ness monster surface. The image of $\rho_S$ is the group of automorphisms of $\mathop{\mathrm{H_1}}\nolimits1(S;\mathop{\mathrm{\mathbb{Z}}}\nolimits)$ that preserve the algebraic intersection form.

Figures (11)

  • Figure 1: Curves for a homology basis of Jacob's ladder
  • Figure 2: A mapping class inducing $-\phi_2$
  • Figure 3: Two pairs of flare surfaces with the same homology, but different spaces of ends
  • Figure 4: Disjoint flare surfaces whose homologies have trivial intersection and the subsurface $K$ in the proof of Lemma \ref{['lem:flaredisjoint']}.
  • Figure 5: The subsurfaces in condition (4)
  • ...and 6 more figures

Theorems & Definitions (66)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 56 more