The Casson invariant and the word metric on the Torelli group

TL;DR

This paper establishes a quadratic bound on the Casson invariant of homology 3-spheres in terms of the word metric distance in the Torelli group, demonstrating the bound's sharpness with constructed examples.

Contribution

It provides the first explicit quadratic bound relating the Casson invariant to the Torelli group's word metric and constructs examples confirming the bound's asymptotic sharpness.

Findings

Casson invariant is bounded quadratically by Torelli group word metric

Constructed examples show the bound is asymptotically sharp

Provides new quantitative link between 3-manifold invariants and mapping class groups

Abstract

We bound the value of the Casson invariant of any integral homology 3-sphere $M$ by a constant times the distance-squared to the identity, measured in any word metric on the Torelli group $\T$, of the element of $\T$ associated to any Heegaard splitting of $M$. We construct examples which show this bound is asymptotically sharp.