The Torelli groups for genus 2 and 3 surfaces

Mess, Geoffrey

doi:10.1016/0040-9383(92)90008-6

Cited by 104 publications

(127 citation statements)

References 23 publications

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“…In 1983, Johnson [29] proved that I(S g ) is finitely generated for g ≥ 3, and in 1986, McCullough-Miller [34] proved that I(S 2 ) is not finitely generated. Mess [37] improved on this in 1992 by proving Theorem D. At the same time, Mess [37] showed that H 3 (I(S 3 ), Z) is not finitely generated (Mess credits the argument to Johnson-Millson). In 2001, Akita [2] proved that H (I(S g ), Z) is not finitely generated for g ≥ 7.…”

Section: Theorem E the Complex B(s Gmentioning

confidence: 99%

The dimension of the Torelli group

Bestvina

Bux

Margalit

2009

J. Amer. Math. Soc.

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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.

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Section: Theorem E the Complex B(s Gmentioning

confidence: 99%

The dimension of the Torelli group

Bestvina

Bux

Margalit

2009

J. Amer. Math. Soc.

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“…Note that we have already proved them in the argument of the previous sections. Since K 2 is a free group [17], the third equality is trivial. The output is −576(a 1 ∧ a 2 ) ⊗ (a 1 ∧ a 2 ).…”

Section: The Cases Of G = 2mentioning

confidence: 93%

“…However, we still do not have enough information on K g . McCullough-Miller [16] showed that K 2 = I 2 is not finitely generated, and Mess [17] showed that it is a free group of infinite rank. Recently, Biss-Farb [5] showed that K g is not finitely generated for all g ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

The second Johnson homomorphism and the second rational cohomology of the Johnson kernel

Sakasai

2007

Math. Proc. Camb. Phil. Soc.

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ABSTRACT. The Johnson kernel is the subgroup of the mapping class group of a surface generated by Dehn twists along bounding simple closed curves, and has the second Johnson homomorphism as a free abelian quotient. In terms of the representation theory of the symplectic group, we give a complete description of cup products of two classes in the first rational cohomology of the Johnson kernel obtained by the rational dual of the second Johnson homomorphism.

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“…Recently Biss and Farb [5] proved that the group K g is not finitely generated for all g ≥ 3 (K 2 is known to be an infinitely generated free group by Mess [69]). However it is still not yet known whether the abelianization H 1 (K g ) is finitely generated or not (cf.…”

Section: Higher Geometry Of the Mapping Class Groupmentioning

confidence: 99%

Cohomological structure of the mapping class group and beyond

Morita¹

2006

Proceedings of Symposia in Pure Mathematics

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Abstract. In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces.Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders.

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The Torelli groups for genus 2 and 3 surfaces

Cited by 104 publications

References 23 publications

The dimension of the Torelli group

The dimension of the Torelli group

The second Johnson homomorphism and the second rational cohomology of the Johnson kernel

Cohomological structure of the mapping class group and beyond

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