Algebraic & Geometric Topology

Lagrangian mapping class groups from a group homological point of view

Takuya Sakasai

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We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3–manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a byproduct of this investigation, we determine the second homology of the mapping class group of a surface of genus 3.

Article information

Algebr. Geom. Topol., Volume 12, Number 1 (2012), 267-291.

Received: 20 November 2010
Revised: 1 November 2011
Accepted: 10 November 2011
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20]
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 57R20: Characteristic classes and numbers

mapping class group Torelli group Lagrangian filtration Miller–Morita–Mumford class


Sakasai, Takuya. Lagrangian mapping class groups from a group homological point of view. Algebr. Geom. Topol. 12 (2012), no. 1, 267--291. doi:10.2140/agt.2012.12.267.

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