The virtual cohomological dimension of the mapping class group of an orientable surface
type: Item
instance of: scholarly article
title: The virtual cohomological dimension of the mapping class group of an orientable surface
zbMATH Open document ID: 0592.57009
DOI: 10.1007/BF01388737
author: J. L. Harer
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143338
review text: Let \(\Gamma =\Gamma^ s_{g,r}\) be the mapping class group of an orientable surface \(F=F^ s_{g,r}\) of genus \(g\) with \(s\) punctures and \(r\) boundary components. Thus each element of \(\Gamma\) is represented by an orientation preserving diffeomorphism of \(F\) which preserves the punctures of \(F\) individually and restricts to the identity on \(\partial F\). The group \(\Gamma\) is known to possess torsion-free subgroups of finite index, i.e. \(\Gamma\) is virtually torsion-free. The main result of the paper under review asserts that \(\Gamma\) has, in addition, cohomological properties parallelling those of the arithmetic groups. Theorem 4.1. If \(2(g-1)+r+s\) is positive, the group \(\Gamma =\Gamma^ s_{g,r}\) is a virtual duality group (in the sense of Bieri-Eckmann) of dimension \(d=d(g,r,s)\) whose dualizing module \(H^ d(\Gamma, {\mathbb Z}\Gamma)\) is free abelian. The cohomological dimension d is \(4(g- 1)+2r+s\) if \(g>0\) and \(r+s>0\), it is 4(g-1)-1 if \(r=s=0\) and \(2r+s-3\) if \(g=0.\) The groups for which \(2(g-1)+r+s\) is not positive are either trivial, infinite cyclic or isomorphic to \(\text{SL}(2,{\mathbb Z})\); in particular, they are virtual duality groups. The main steps in the proof of the theorem are the following ones: First assume there is a boundary component. By plugging a boundary component with a punctured disc one obtains an embedding \(F^ s_{g,r}\hookrightarrow F^{s+1}_{g,r-1}\) which induces a projection of \(\Gamma^ s_{g,r}\) onto \(\Gamma^{s+1}_{g,r-1}\) with infinite cyclic kernel. This fact allows one to concentrate on the cases with \(r=0.\) If \(r=0\) the group \(\Gamma^ s_ g=\Gamma^ s_{g,0}\) acts properly discontinuously on Teichmüller space \(T=T^ s_ g\) and the stabilizers of points are all finite. The space \(T\) is known to be homeomorphic to Euclidean space of dimension \(6(g-1)+2s\) whence each torsion-free subgroup of \(\Gamma^ s_ g\) has cohomological dimension bounded by \(6(g-1)+2s\). To obtain sharper resu…
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 30F30
zbMATH DE Number: 3951591
zbMATH Keywords: mapping class group of an orientable surface
zbMATH Keywords: virtually torsion-free
zbMATH Keywords: arithmetic groups
zbMATH Keywords: virtual duality group
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Borel- Serre bordification
zbMATH Keywords: quadratic differential
Wikidata QID: Q56519378
MaRDI profile type: Publication
cites work: Deformation retracts with lowest possible dimension of arithmetic quotients of self-adjoint homogeneous cones
cites work: Groups with homological duality generalizing Poincaré duality
cites work: Q3708977
cites work: Corners and arithmetic groups (Appendice: Arrondissement des varietes a coins par A. Douady et L. Herault)
cites work: On the Lusternik-Schnirelmann category of abstract groups
cites work: The second homology group of the mapping class group of an orientable surface
cites work: On the Kodaira dimension of the moduli space of curves
cites work: Quadratic differentials and foliations
cites work: Arithmeticity of the irreducible lattices in the semi-simple groups of rank greater than 1
cites work: Characteristic classes of surface bundles
cites work: Abelian quotients of the Teichmüller modular group
cites work: Two Theorems on the Mapping Class Group of a Surface
cites work: On quadratic differentials with closed trajectories and second order poles
cites work: Q3338499
cites work: The cohomology of \(SL_3(Z)\)
cites work: Finiteness conditions for CW-complexes. I, II.
cites work: On the homology of the moduli space of stable curves
description: scientific article; zbMATH DE number 3951591