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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The dimension of the Torelli group
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by Mladen Bestvina, Kai-Uwe Bux and Dan Margalit PDF
J. Amer. Math. Soc. 23 (2010), 61-105 Request permission

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
  • MR Author ID: 36095
  • 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
  • MR Author ID: 706322
  • Email: dan.margalit@tufts.edu
  • 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.
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 23 (2010), 61-105
  • MSC (2000): Primary 20F34; Secondary 57M07
  • DOI: https://doi.org/10.1090/S0894-0347-09-00643-2
  • MathSciNet review: 2552249