2013/11/29 by Joan S. Birman, Joan Birman, Nathan Broaddus +5
Computer Science · Mathematics · #20F38 #57M60 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GT #msc:20F38 #msc:57M60
paper · pdf · doi:10.48550/arxiv.1311.7646
21 pages, 7 figures; Section 4 revised along with minor corrections throughout
openalex publication_date 2013/11/29 · arxiv created 2014/06/09 · arxiv updated 2014/06/10 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Aramayona and Leininger have provided a "finite rigid subset"\n mathfrakX(\Σ) of the curve complex mathscrC(\Σ) of a surface\n\Σ = \Σng, characterized by the fact that any simplicial injection\n mathfrakX(\Σ) \→ mathscrC(\Σ) is induced by a unique element\nof the mapping class group \Mod(\Σ). In this paper we prove that,\nin the case of the sphere with n\≥ 5 marked points, the reduced homology\nclass of the finite rigid set of Aramayona and Leininger is a\n\Mod(\Σ)-module generator for the reduced homology of the curve\ncomplex mathscrC(\Σ), answering in the affirmative a question posed by\nAramayona and Leininger. For the surface \Σ = \Σgn with g\≥ 3\nand n\∈ 0,1 we find that the finite rigid set mathfrakX(\Σ) of\nAramayona and Leininger contains a proper subcomplex X(\Σ) whose reduced\nhomology class is a \Mod(\Σ)-module generator for the reduced\nhomology of mathscrC(\Σ) but which is not itself rigid.\n