Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The normal closure of a bounding pair map in its mapping class group

Lei Chen, Weiyan Chen, Justin Lanier

TL;DR

The paper analyzes the normal closure in the mapping class group generated by genus $n$ bounding pair maps, generalizing Johnson’s result for genus $1$. It provides two descriptions via the Chillingworth congruence and the Casson–Morita invariant, showing $ ext{W}_g^1(n)=[ ext{Ch}_g^1[2n], ext{M}_g^1]$ and $ ext{W}_g^1(n)= ext{ker}(d_{4n})$, with explicit index and parity-dependent formulas. The methods combine Johnson's homomorphism, Chillingworth theory, lantern relations, and Putman’s connectivity lemma to control generators (bounding pairs and separating twists) and to relate these subgroups to the Johnson kernel and Torelli subgroups. The results extend to punctured and closed surfaces via Birman exact sequences, yielding a comprehensive picture of how genus-$n$ bounding pair maps generate normal closures and their homological implications across surface types.

Abstract

Johnson showed that the normal subgroup of a mapping class group generated by the genus $1$ bounding pair maps is equal to the Torelli group. Generalizing Johnson's result, we give two descriptions of the normal subgroup generated by the genus $n$ bounding pair maps using the Chillingworth and the Casson--Morita invariants.

The normal closure of a bounding pair map in its mapping class group

TL;DR

The paper analyzes the normal closure in the mapping class group generated by genus bounding pair maps, generalizing Johnson’s result for genus . It provides two descriptions via the Chillingworth congruence and the Casson–Morita invariant, showing and , with explicit index and parity-dependent formulas. The methods combine Johnson's homomorphism, Chillingworth theory, lantern relations, and Putman’s connectivity lemma to control generators (bounding pairs and separating twists) and to relate these subgroups to the Johnson kernel and Torelli subgroups. The results extend to punctured and closed surfaces via Birman exact sequences, yielding a comprehensive picture of how genus- bounding pair maps generate normal closures and their homological implications across surface types.

Abstract

Johnson showed that the normal subgroup of a mapping class group generated by the genus bounding pair maps is equal to the Torelli group. Generalizing Johnson's result, we give two descriptions of the normal subgroup generated by the genus bounding pair maps using the Chillingworth and the Casson--Morita invariants.
Paper Structure (15 sections, 29 theorems, 79 equations, 8 figures)

This paper contains 15 sections, 29 theorems, 79 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Image of $\mathcal{W}_g^1(n)$ under the Johnson homomorphism
  3. Chillingworth and Johnson homomorphisms
  4. Calculating $\tau_g^1(\mathcal{W}_g^1(n))$
  5. Separating twists in $\mathcal{W}(n)$
  6. Proving the main results
  7. The Casson--Morita map and its restriction to $\mathop{\mathrm{Ch}}\nolimits_g^1[2n]$
  8. Proof of Theorem \ref{['thm:W is Ch M commutator']}
  9. Proof of Theorem \ref{['thm: W is kerd']} and Corollary \ref{['cor:W vs K']}
  10. Proof of Corollary \ref{['cor:homological results']}
  11. Image of $d$ on various subgroups of Torelli groups
  12. Punctured and closed surfaces
  13. Point-pushing subgroups and genus $g-1$ bounding pair maps
  14. $\mathcal{W}(n)$ for once-punctured surfaces
  15. $\mathcal{W}(n)$ for closed surfaces

Key Result

Theorem 1.1

When $1\le n\le g-2$, we have that

Figures (8)

  • Figure 1: $T_aT_b^{-1}$ is a genus 2 bounding pair map.
  • Figure 2: Two lantern configurations used to show that $\mathcal{T}(n)$ and $\mathcal{T}(n+1)$ respectively are subgroups of $\mathcal{W}(n)$. The asterisks indicate the boundary components, and the boxed values in the subsurfaces indicate their genus.
  • Figure 3: Left: the lantern configuration used to show that $\mathcal{T}(1)$ is a subgroup of $\mathcal{W}(n)$. Right: the lantern configuration used to prove Proposition \ref{['prop:nested']}. The asterisks indicate the boundary components, and the boxed values in the subsurfaces indicate their genus.
  • Figure 4: Lickorish's generating set from Figure 5.7 in primer with the curve $\alpha$ added.
  • Figure 5: $B_0=T_aT_b^{-1}$ is called a homological genus 0 bounding pair map in Chilling.
  • ...and 3 more figures

Theorems & Definitions (51)

  • Theorem 1.1: First description of $\mathbf{\mathcal{W}_g^1(n)}$
  • Theorem 1.2: Second description of $\mathbf{\mathcal{W}_g^1(n)}$
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 41 more