The dimension of the Torelli group

Bestvina, Mladen¹ ; Bux, Kai-Uwe² ; Margalit, Dan³

¹ Department of Mathematics, University of Utah, 155 S 1400 East, Salt Lake City, Utah 84112-0090
² Department of Mathematics, University of Virginia, Kerchof Hall 229, Charlottesville, Virginia 22903-4137
³ Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, Massachusetts 02155

Journal of the American Mathematical Society, Tome 23 (2010) no. 1, pp. 61-105

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-09-00643-2

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