Finite rigid sets and homologically nontrivial spheres in the curve complex of a surface. (English) Zbl 1308.57009

Birman, Joan; Broaddus, Nathan; Menasco, William

J. Topol. Anal. 7, No. 1, 47-71 (2015).

Let \(\text{Mod}(\Sigma_g^n)\) denote the mapping class group of a surface \(\Sigma = \Sigma_g^n\) of genus \(g\) with \(n\) marked points (allowing permutations of the marked points). J. Aramayona and C. J. Leininger [J. Topol. Anal. 5, No. 2, 183–203 (2013; Zbl 1277.57017)] constructed a finite “rigid” subset \({\mathfrak X} (\Sigma)\) of the curve complex \({\mathcal C}(\Sigma)\) of the surface, i.e. with the property that every simplicial embedding of \({\mathfrak X} (\Sigma)\) into \({\mathcal C}(\Sigma)\) is induced by a unique element of the mapping class group \(\text{Mod}(\Sigma)\) (in its action on \({\mathcal C}(\Sigma)\)). Answering a question of Aramayona and Leininger, it is proved in the present paper that the reduced homology class of the \((n-4)\)-sphere \({\mathfrak X} (\Sigma)\) in \({\mathcal C}(\Sigma)\) is a \(\text{Mod}(\Sigma)\)-module generator of the reduced homology of the curve complex. For a surface of genus \(g \geq 3\) and \(n=0\) or 1, it is shown that the finite rigid set \({\mathfrak X} (\Sigma)\) contains a proper subcomplex (a \((2g-2)\)-sphere) whose reduced homology class is a \(\text{Mod}(\Sigma)\)-module generator of the reduced homology of the curve complex but which is not itself rigid.

Reviewer: Bruno Zimmermann (Trieste)

MSC:

57M99

General low-dimensional topology

Keywords:

surface; curve complex; mapping class group

Citations:

Zbl 1277.57017

References:

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