The dimension of the Torelli group

Abstract

We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus g ≥ 2 g \geq 2 is equal to 3 g − 5 3g-5 . This answers a question of Mess, who proved the lower bound and settled the case of g = 2 g=2 . We also find the cohomological dimension of the Johnson kernel (the subgroup of the Torelli group generated by Dehn twists about separating curves) to be 2 g − 3 2g-3 . For g ≥ 2 g \geq 2 , we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli group in genus 2. The main tool is a new contractible complex, called the “complex of minimizing cycles”, on which the Torelli group acts.