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On the Birman exact sequence of the subgroups of the mapping class group of genus three

Ma Luo, Tatsunari Watanabe

TL;DR

This work analyzes the nonsplitting of profinite and graded Birman exact sequences associated with subgroups of the mapping class group in genus $g\ge 3$. Using relative (unipotent) completion together with mixed Hodge theory and $\operatorname{Sp}(H)$-representation theory, the authors derive obstructions that prevent the existence of sections in both the profinite Birman sequence for finite-index subgroups containing the Johnson subgroup and the graded Lie algebra version for Torelli groups, extending prior results from higher genus to $g=3$. A key technical input is the interaction between hyperelliptic and non-hyperelliptic cases via the relative completions ${\mathcal G}_{g,n}$, ${\mathcal D}_{g,n}$ and their unipotent radicals, as well as explicit weight-graded obstructions coming from $\Lambda^2(\Lambda_0^3H)$ containing $\Lambda_0^2H$. The results imply that the classical Birman exact sequence does not split for these subgroups, highlighting deep structural constraints in the arithmetic and geometric topology of moduli spaces and their Torelli-type subgroups.

Abstract

We prove that for any finite index subgroup of the mapping class group containing the Johnson subgroup, the profinite Birman exact sequence does not split in genus $g\ge 3$, extending prior results of Hain and the second author for $g\ge 4$. For the Torelli group, we prove that the graded Lie algebra version of the Birman exact sequence admits no section with symplectic equivariance, extending Hain's result from $g\ge 4$ to $g=3$. These results are deduced by our main tool, relative completion, with the help of Hodge theory and representation theory of symplectic groups, along with explicit structural obstructions coming from hyperelliptic mapping class groups.

On the Birman exact sequence of the subgroups of the mapping class group of genus three

TL;DR

This work analyzes the nonsplitting of profinite and graded Birman exact sequences associated with subgroups of the mapping class group in genus . Using relative (unipotent) completion together with mixed Hodge theory and -representation theory, the authors derive obstructions that prevent the existence of sections in both the profinite Birman sequence for finite-index subgroups containing the Johnson subgroup and the graded Lie algebra version for Torelli groups, extending prior results from higher genus to . A key technical input is the interaction between hyperelliptic and non-hyperelliptic cases via the relative completions , and their unipotent radicals, as well as explicit weight-graded obstructions coming from containing . The results imply that the classical Birman exact sequence does not split for these subgroups, highlighting deep structural constraints in the arithmetic and geometric topology of moduli spaces and their Torelli-type subgroups.

Abstract

We prove that for any finite index subgroup of the mapping class group containing the Johnson subgroup, the profinite Birman exact sequence does not split in genus , extending prior results of Hain and the second author for . For the Torelli group, we prove that the graded Lie algebra version of the Birman exact sequence admits no section with symplectic equivariance, extending Hain's result from to . These results are deduced by our main tool, relative completion, with the help of Hodge theory and representation theory of symplectic groups, along with explicit structural obstructions coming from hyperelliptic mapping class groups.

Paper Structure

This paper contains 25 sections, 24 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. The mapping class groups and hyperelliptic mapping class groups
  3. Symplectic representation of the mapping class group
  4. Level structure
  5. Hyperelliptic Torelli group
  6. The universal curves and Torelli space
  7. Moduli space of hyperelliptic curves
  8. Universal curves
  9. Moduli stack of curves
  10. Moduli stack of hyperelliptic curves
  11. The Malcev completion of groups
  12. Malcev completion and Malcev Lie algebra
  13. Continuous unipotent completion
  14. Relative completion
  15. Relative completion of a discrete group
  16. ...and 10 more sections

Key Result

Theorem 1

Suppose that ${\Gamma}$ is a finite-index subgroup of ${\Gamma}_g$ containing $K_g$. If $g\geq 3$, the profinite Birman exact sequence (profinite birman for subgroup) for $\widehat{{\Gamma}}$ does not split. Consequently, the Birman exact sequence (birman for subgroup) for ${\Gamma}$ does not split

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 2.1: Birman-Hilden birman-hilden
  • Example 4.1: The unipotent completion of Torelli groups
  • Example 4.2: The unipotent completion of a fundamental group
  • Theorem 4.3
  • Remark 4.4
  • Example 4.5
  • Example 4.6
  • ...and 33 more