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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

The dimension of the Torelli group


Authors: Mladen Bestvina, Kai-Uwe Bux and Dan Margalit
Journal: J. Amer. Math. Soc. 23 (2010), 61-105
MSC (2000): Primary 20F34; Secondary 57M07
Published electronically: July 10, 2009
MathSciNet review: 2552249
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Abstract: 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.


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Additional Information

Mladen Bestvina
Affiliation: Department of Mathematics, University of Utah, 155 S 1400 East, Salt Lake City, Utah 84112-0090
Email: bestvina@math.utah.edu

Kai-Uwe Bux
Affiliation: Department of Mathematics, University of Virginia, Kerchof Hall 229, Charlottesville, Virginia 22903-4137
Email: kb2ue@virginia.edu

Dan Margalit
Affiliation: Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, Massachusetts 02155
Email: dan.margalit@tufts.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-09-00643-2
PII: S 0894-0347(09)00643-2
Keywords: Mapping class group, Torelli group, Johnson kernel, cohomological dimension, complex of minimizing cycles
Received by editor(s): September 7, 2007
Published electronically: July 10, 2009
Additional Notes: The first and third authors gratefully acknowledge support by the National Science Foundation.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.