On finite simple groups acting on homology spheres with small fixed\n point sets

2014/05/28 by Bruno Zimmermann, Zimmermann, Bruno P.
Mathematics · #57M60 #57S17 #57S25 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1405.7220

openalex publication_date 2014/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A finite nonabelian simple group does not admit a free action on a homology\nsphere, and the only finite simple group which acts on a homology sphere with\nat most 0-dimensional fixed point sets ("pseudofree action") is the alternating\ngroup A₅ acting on the 2-sphere. Our first main theorem is the finiteness\nresult that there are only finitely many finite simple groups which admit a\nsmooth action on a homology sphere with at most d-dimensional fixed points\nsets, for a fixed d. We then go on proving that the finite simple groups acting\non a homology sphere with at most 1-dimensional fixed point sets are the\nalternating group A₅ in dimensions 2, 3 and 5, the linear fractional group\nPSL₂(7) in dimension 5, and possibly the unitary group PSU₃(3) in dimension 5\n(we conjecture that it does not admit any action on a homology 5-sphere but\ncannot exclude it at present). Finally, we discuss the situation for arbitrary\nfinite groups which admit an action on a homology 3-sphere.\n

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