On finite simple and nonsolvable groups acting on closed 4-manifolds

2008/03/31 by Mattia Mecchia, Mecchia, Mattia, Bruno Zimmermann +1
Mathematics · #57M60 #57S17 #57S25 #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0803.4454

openalex publication_date 2008/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the only finite nonabelian simple groups which admit a locally linear, homologically trivial action on a closed simply connected 4-manifold M (or on a 4-manifold with trivial first homology) are the alternating groups A₅, A₆ and the linear fractional group PSL(2,7) (we note that for homologically nontrivial actions all finite groups occur). The situation depends strongly on the second Betti number b₂(M) of M and has been known before if b₂(M) is different from two, so the main new result of the paper concerns the case b₂(M)=2. We prove that the only simple group that occurs in this case is A₅, and then give a short list of finite nonsolvable groups which contains all candidates for actions of such groups.

Related

• Homologically Trivial Group Actions on 4-Manifolds
• On finite simple groups acting on homology spheres with small fixed\n point sets
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• Symmetry groups of non-simply-connected four-manifolds
• On finite groups acting on acyclic low-dimensional manifolds