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

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 A 5 , A 6 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 b 2 (M) of M and was known before if b 2 (M) is different from two, so the main new result concerns the case b 2 (M) = 2. We prove that the only simple group that occurs in this case is A 5 , and then deduce a short list of finite nonsolvable groups which contains all candidates for actions of such groups.