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The normal closure of a homological genus 0 bounding pair map

Lei Chen, Weiyan Chen

TL;DR

The paper extends the normal-generation framework for bounding-pair maps to the genus-0 (homological) case by identifying the kernel of the Casson–Morita $d$-map on the Chillingworth subgroup ${\text{Ch}_{g}^{1}}$ with the commutator ${[\text{Ch}_{g}^{1}}, {\mathcal{M}_g^1]}$, proving this kernel is normally generated by a single homological genus-0 BP map $B_0$. It introduces a homological genus for fake bounding pairs ${g^{\alpha}}(a,b)$ via the Chillingworth map, classifies genus-0 cases, and uses lantern relations to realize $B_0$ and related elements as commutators. The work then carries out a detailed cohomological analysis of ${\text{Ch}_{g}^{1}}$ and the subgroup ${\mathcal{W}_g^1}(0)$, establishing invariant parts of $H^1$ and $H_1$ with respect to ${\mathcal{M}_g^1}$ and showing that ${\text{Ch}_{g}^{1}}$ is normally generated by a single element $H_0$. A reflexive use of curve complexes, especially $C_0^{\alpha}$, together with Putman’s connectivity lemma, yields path-connectedness and transitivity results that underpin the main normal-generation claims.

Abstract

Justin Lanier and the authors recently determined the group normally generated by a single bounding pair map of genus $n$. We related this subgroup with the Chillingworth subgroup and the Casson--Morita's $d$ map. In this paper, we extend the results to the case when $n=0$. Let $\mathcal{M}_g^1$ be the mapping class group, $\text{Ch}_g^1$ be the Chillingworth subgroup and $d$ be the Casson--Morita's $d$-map. We show that $\text{Ker}(d)=[\text{Ch}_g^1,\mathcal{M}_g^1]$ and it is generated by a single homological genus 0 bounding pair map. We also construct an element $H_0\in \text{Ch}_g^1$, and show that $\text{Ch}_g^1$ is normally generated by this single element $H_0$.

The normal closure of a homological genus 0 bounding pair map

TL;DR

The paper extends the normal-generation framework for bounding-pair maps to the genus-0 (homological) case by identifying the kernel of the Casson–Morita -map on the Chillingworth subgroup with the commutator , proving this kernel is normally generated by a single homological genus-0 BP map . It introduces a homological genus for fake bounding pairs via the Chillingworth map, classifies genus-0 cases, and uses lantern relations to realize and related elements as commutators. The work then carries out a detailed cohomological analysis of and the subgroup , establishing invariant parts of and with respect to and showing that is normally generated by a single element . A reflexive use of curve complexes, especially , together with Putman’s connectivity lemma, yields path-connectedness and transitivity results that underpin the main normal-generation claims.

Abstract

Justin Lanier and the authors recently determined the group normally generated by a single bounding pair map of genus . We related this subgroup with the Chillingworth subgroup and the Casson--Morita's map. In this paper, we extend the results to the case when . Let be the mapping class group, be the Chillingworth subgroup and be the Casson--Morita's -map. We show that and it is generated by a single homological genus 0 bounding pair map. We also construct an element , and show that is normally generated by this single element .

Paper Structure

This paper contains 18 sections, 25 theorems, 82 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The Chillingworth homomorphism and the Chillingworth subgroup
  3. The Chillingworth homomorphism and the Chillingworth subgroup
  4. Homological genus of a fake bounding pair map
  5. Classification of homological genus 0 bounding pair maps
  6. The proof of that $B_0\in [{\text{Ch}{_g^1}},{\mathcal{M}_g^1}]$
  7. The proof of that ${\mathcal{W}_g^1}(0)= [{\mathcal{W}_g^1}(0),{\mathcal{M}_g^1}]$
  8. The proof of that $T_1^2\in [{\text{Ch}{_g^1}},{\mathcal{M}_g^1}]$
  9. Calculations about $Ch_g^1$
  10. The cohomology class of intersection form
  11. The computation of $H^1({\text{Ch}{_g^1}};\mathbb Z/m\mathbb Z)^{{\mathcal{M}_g^1}}$
  12. The proof of that $T_1\in [{\text{Ch}{_g^1}},{\mathcal{M}_g^1}]$.
  13. The proof of the right equality of Theorem \ref{['main1']}
  14. The proof of Theorem \ref{['thm:homological results']}
  15. The curve complex $C_0^\alpha$ and the proof of Theorem \ref{['C_0']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

When $1\le n\le g-2$, the normal subgroup of ${\mathcal{M}_g^1}$ generated by a genus $n$ bounding pair map $BP_n$ is

Figures (11)

  • Figure 1: A homological genus 0 bounding pair map $B_0=T_aT_b^{-1}$.
  • Figure 2: The number $g-3$ denotes the genus of the subsurface.
  • Figure 3: The boxed value $m$ denotes the genus of the subsurface.
  • Figure 4: Left: A small neighborhood of $b\cup c$ forms a lantern. $A,B,C,D$ denote the four arcs such that $b=AB$ and $c=CD$ with orientations. Right: A different picture of the same lantern $N$.
  • Figure 5: Left: the purple curve $e$ intersects $AC^{-1}$ and $DB^{-1}$ once but with opposite signs. This is cannot happen. Right: the three subsurfaces $\Sigma_1, \Sigma_2, \Sigma_3$ outside of $N$.
  • ...and 6 more figures

Theorems & Definitions (55)

  • Theorem 1.1: Theorem 1.1 and 1.2 in Normal
  • Theorem 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7: ${\text{Ch}{_g^1}}$–orbits of nonseparating curves
  • Lemma 2.1
  • Proposition 2.2: homological genus of a fake bounding pair
  • Definition 2.3: homology curve complex
  • Theorem 2.4: Putman, Theorem 1.9 in Putman
  • ...and 45 more