The second rational homology of the Torelli group is finitely generated

TL;DR

This paper proves that the second rational homology of the Torelli group for high-genus surfaces is finite dimensional, advancing understanding of its algebraic structure and addressing a longstanding question.

Contribution

It establishes the finite dimensionality of the second rational homology for the Torelli group when genus is at least 51, a significant step in understanding its algebraic properties.

Findings

Second rational homology is finite dimensional for genus ≥ 51

Rules out the simplest obstruction to finite presentability

Provides partial answer to a question of Bestvina

Abstract

We prove that second rational homology of the Torelli group of an orientable closed surface of genus g is finite dimensional for g at least 51. This rules out the simplest obstruction to the Torelli group being finitely presented and provides a partial answer to a question of Bestvina.