[Paper Review] The Birman exact sequence does not virtually split

Lei Chen, Nick Salter|arXiv (Cornell University)|Apr 30, 2018

Finite Group Theory Research2 references3 citations

TL;DR

This paper proves that the Birman exact sequence for mapping class groups of surfaces of genus $g \geq 4$ does not virtually split, meaning no finite-index subgroup of the Torelli group admits a section. Using homological constraints and a detailed analysis of a candidate section map $s$, the authors derive a contradiction by showing that the induced map on cohomology cannot vanish on the diagonal class, thus resolving a long-standing question originally claimed (but incorrectly) by G. Mess in 1990.

ABSTRACT

This paper answers a basic question about the Birman exact sequence in the theory of mapping class groups. We prove that the Birman exact sequence does not admit a section over any subgroup $Γ$ contained in the Torelli group with finite index. A fortiori this proves that there is no section of the Birman exact sequence for any finite-index subgroup of the full mapping class group. This theorem was announced in a 1990 preprint of G. Mess, but an error was uncovered and described in a recent paper of the first author.

Motivation & Objective

• To resolve the open question of whether the Birman exact sequence virtually splits for mapping class groups of surfaces of genus $g \geq 4$.
• To correct and complete the flawed argument of G. Mess (1990) which claimed the same result but contained a critical error in subgroup assumptions.
• To show that no finite-index subgroup of the Torelli group admits a section of the Birman exact sequence.
• To establish that the universal family of homologically framed curves over Torelli space does not admit any continuous multisection.
• To demonstrate that the absence of a section persists even in the virtual sense, despite the existence of multisections in genus 2.

Proposed method

• Construct a candidate section map $s: \overline{\mathcal{H}} \to \pi_1(\Sigma_p)$ from a hypothetical section of the Birman exact sequence.
• Analyze the image of $s$ using the action of Dehn twists and conjugacy invariance, showing that $s(x)$ must be trivial for all $x$ in the image of the Torelli group.
• Use the change-of-coordinates principle to show that any nontrivial element in $\pi_1(\Sigma_p)$ must intersect some separating curve, leading to a contradiction if $s(x)$ is nontrivial.
• Apply Poincaré duality and Thom isomorphism to show that the pullback of the diagonal class $[\Delta]$ under $i \times s$ vanishes in cohomology.
• Derive a contradiction by showing that the pullback of $[\Delta]$ is nonzero in $H^2(\overline{\mathcal{H}})$, contradicting the vanishing of $\iota^*([\Delta])$ in $H^2(\operatorname{PConf}_2(\Sigma_p))$.
• Distinguish two cases: (A) where $s^* = i^*$ in degree 1, and (B) where $s^* = 0$ in positive degrees, showing both lead to contradiction via nonvanishing Euler characteristic terms.

Experimental results

Research questions

• RQ1 Does the Birman exact sequence virtually split for any finite-index subgroup of the Torelli group in genus $g \geq 4$?
• RQ2 Can a continuous multisection be constructed for the universal family of homologically framed curves over Torelli space in genus $g \geq 4$?
• RQ3 Does the flawed 1990 argument by G. Mess correctly establish that the Birman sequence does not virtually split?
• RQ4 Is there a homological obstruction to the existence of a section over any finite-sheeted cover of the moduli space $\mathcal{M}_g$ for $g \geq 4$?
• RQ5 Can the existence of a section be ruled out even when allowing for multisections (i.e., continuous choices of multiple distinct points)?

Key findings

• The Birman exact sequence does not virtually split for any finite-index subgroup $\Gamma \leq \mathcal{I}(\Sigma_g)$ when $g \geq 4$, resolving a long-standing open problem.
• The homomorphism $s$ constructed from a hypothetical section must map all elements to the trivial element in $\pi_1(\Sigma_p)$, contradicting the nontriviality of the cohomological pullback.
• The pullback of the diagonal class $[\Delta]$ under $i \times s$ is nonzero in $H^2(\overline{\mathcal{H}})$, contradicting the fact that $\iota^*([\Delta]) = 0$ in $H^2(\operatorname{PConf}_2(\Sigma_p))$.
• In Case (A), where $s^* = i^*$ in degree 1, the pullback of $[\Sigma_p]$ is $\chi(\Sigma_p)[\overline{\mathcal{H}}]$, which is nonzero and thus contradicts the vanishing condition.
• In Case (B), where $s^* = 0$ in positive degrees, the pullback of $[\Sigma_p]$ is still $\chi(\Sigma_p)[\overline{\mathcal{H}}]$, which is nonzero and again contradicts the required vanishing.
• The result implies that the universal family $\mathscr{I}_{g,*} \to \mathscr{I}_g$ does not admit any continuous multisection for $g \geq 4$, extending the non-splitting result to the multisection setting.