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

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.

MSC:

57M99 General low-dimensional topology

Citations:

Zbl 1277.57017
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References:

[1] DOI: 10.1142/S1793525313500076 · Zbl 1277.57017
[2] DOI: 10.1215/00127094-1645634 · Zbl 1250.57032
[3] DOI: 10.1007/978-1-4684-9327-6
[4] DOI: 10.1016/j.topol.2005.08.010 · Zbl 1099.52001
[5] DOI: 10.1007/BF01388737 · Zbl 0592.57009
[6] DOI: 10.1016/0166-8641(91)90050-V · Zbl 0727.57012
[7] DOI: 10.1155/S1073792897000433 · Zbl 0890.57018
[8] DOI: 10.1515/CRELLE.2009.087 · Zbl 1190.57014
[9] DOI: 10.1016/S0040-9383(99)00008-7 · Zbl 0951.32012
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