The cohomological dimension of the terms of the Johnson filtration

TL;DR

This paper determines the cohomological dimension of the higher terms in the Johnson filtration for closed orientable surfaces, answering a longstanding question in geometric topology.

Contribution

It establishes that the $k$th term of the Johnson filtration has cohomological dimension $2g - 3$ for all $k \, \geq 3$, $g \, \geq 2$, providing a definitive answer to a question posed by Farb and others.

Findings

Cohomological dimension of Johnson filtration terms is $2g - 3$ for $k \geq 3$, $g \geq 2$.

Answers a question posed by Farb and Bux--Margalit.

Advances understanding of the algebraic and geometric properties of surface mapping class groups.

Abstract

We prove that the $k$th term of the Johnson filtration of a closed, orientable surface of genus $g \geq 2$ has cohomological dimension $2g - 3$ for all $k \geq 3$ and $g \geq 2$. This answers a question of Farb and Bestvina--Bux--Margalit.