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

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

2008-03-31

arxiv.org/abs/0803.4454v1

Mathematics

Geometric Topology

17 pages

Scientific paper

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 give a short list of finite nonsolvable groups which contains all candidates for actions of such groups.

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