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

Mattia Mecchia and Bruno Zimmermann

Vol. 243 (2009), No. 2, 357–374

DOI: 10.2140/pjm.2009.243.357

Abstract

We show that the only finite nonabelian simple groups to 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 was known before if b₂(M) is different from two, so the main new result concerns the case b₂(M) = 2. We prove that the only simple group that occurs in this case is 𝔸₅, and then deduce a short list of finite nonsolvable groups which contains all candidates for actions of such groups.

Keywords

finite group action, simply connected 4-manifold, simple group

Mathematical Subject Classification 2000

Primary: 57M60, 57S17, 57S25

Milestones

Received: 14 April 2008

Revised: 2 February 2009

Accepted: 1 May 2009

Published: 1 December 2009