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 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 has been known before if b_2(M) is different from two, so the main new result of the paper 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.
On finite simple and nonsolvable groups acting on closed 4-manifolds
MECCHIA, MATTIA;ZIMMERMANN, BRUNO
2009-01-01
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 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 has been known before if b_2(M) is different from two, so the main new result of the paper 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.Pubblicazioni consigliate
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