## The virtual cohomological dimension of the mapping class group of an orientable surface.(English)Zbl 0592.57009

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 results a contractible bordification $$\bar T$$ of $$T$$ with compact quotient $$\bar T/\Gamma$$ is called for. A Borel-Serre bordification of $$T^ 0_ g$$ has been constructed by W. Harvey using analytical methods [Ann. Math. Stud. 97, 241–251 (1981; Zbl 0461.30036)].
In this paper such a bordification $$W$$ is built for the spaces $$T^ s_ g$$ with $$s>0$$ by a different, combinatorial method, and its boundary $$\partial W$$ is shown to be homotopy equivalent to a wedge of spheres of dimension $$2(g-1)+s-1$$ if $$g>0$$, and s-4 if $$g=0$$, $$s\geq 4$$. (The group $$\Gamma^ s_ 0$$ is trivial for $$s<4.)$$ The properties of $$W$$, in conjunction with known results from algebraic topology and cohomology theory of groups, imply the claim of Theorem 4.1 for the cases where $$s>0$$. To establish the claim for $$g>1$$, $$s=0$$ the author makes use of the fact that the inclusion $$F^ 1_{g,0}\hookrightarrow F^ 0_{g,0}=F_ g$$ induces a surjection of $$\Gamma^ 1_ g$$ onto $$\Gamma^ 0_ g$$ whose kernel is $$\pi_ 1(F_ g)$$. Since $$\Gamma^ 0_ g$$ is known to be virtually of finite cohomological dimension, this implies that $$\text{vcd}(\Gamma^ 0_ g)=\text{vcd}(\Gamma^ 1_ g)-2=4(g-1)-1$$. To prove that $$\Gamma^ 0_ g$$ is a duality group he takes recourse to Harvey’s construction of a bordification of $$T^ 0_ g$$ mentioned above. (Alternatively, one could quote a theorem of R. Bieri [J. Pure Appl. Algebra 7, 35–51 (1976; Zbl 0322.20024)] asserting that a quotient of a duality group by a group of type FP is a duality group, provided the dimension of the quotient is finite.)
In addition to the bordification the author describes a spine $$Y$$ of $$T$$, a cell complex of dimension $$4(g-1)+s$$ onto which $$T$$ can be retracted $$\Gamma$$-equivariantly. $$Y$$ has the lowest possible dimension in case $$g>0$$ and $$s>0.$$
The construction of W and Y both make use of an ideal triangulation of $$T^ s_{g,r}$$. This triangulation relies on a theorem of K. Strebel [J. Anal. Math. 19, 373–382 (1967; Zbl 0158.324)] according to which each point of $$T^ s_{g,r}$$ corresponds uniquely to a quadratic differential on $$F^ s_{g,r}$$ with special behaviour near punctures and boundary components.
Reviewer: R. Strebel

### MSC:

 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 20F38 Other groups related to topology or analysis 20J05 Homological methods in group theory 30F30 Differentials on Riemann surfaces

### Citations:

Zbl 0461.30036; Zbl 0322.20024; Zbl 0158.324
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### References:

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