The dimension of the Torelli group

We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus g \geq 2 is equal to 3g-5. This answers a question of Mess, who proved the lower bound and settled the case of g=2. We also find the cohomological dimension of the Johnson kernel (the...

Full description

Saved in:
Bibliographic Details
Published in:Journal of the American Mathematical Society 2010-01, Vol.23 (1), p.61-105
Main Authors: BESTVINA, MLADEN, BUX, KAI-UWE, MARGALIT, DAN
Format: Article
Language:English
Subjects:
Algebra
Closed curves
Commutators
Coordinate systems
Geometric topology
Mathematical surfaces
Mathematical theorems
Vertices
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus g \geq 2 is equal to 3g-5. This answers a question of Mess, who proved the lower bound and settled the case of 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 2g-3. For 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.
ISSN:0894-0347
1088-6834
DOI:10.1090/S0894-0347-09-00643-2