On finite simple and nonsolvable groups acting on homology 4-spheres

Details On finite simple and nonsolvable groups acting on homology 4-spheres On finite simple and nonsolvable groups acting on homology 4-spheres

2005-07-08

arxiv.org/abs/math/0507173v1

Topology Appl. 153 (2006), 2933-2942.

Mathematics

Geometric Topology

15 pages

Scientific paper

The only finite nonabelian simple group acting on a homology 3-sphere - necessarily non-freely - is the dodecahedral group $\Bbb A_5 \cong {\rm PSL}(2,5)$ (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group $\Bbb A_5^* \cong {\rm SL}(2,5)$). In the present paper we show that the only finite simple groups acting on a homology 4-sphere, and in particular on the 4-sphere, are the alternating or linear fractional groups groups $\Bbb A_5 \cong {\rm PSL}(2,5)$ and $\Bbb A_6 \cong {\rm PSL}(2,9)$. From this we deduce a short list of groups which contains all finite nonsolvable groups admitting an action on a homology 4-spheres.

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