Group actions on spheres with rank one isotropy

We show that a rank two finite group G admits a finite G-CW-complex X homotopy equivalent to a sphere, with rank one prime power isotropy, if and only if G does not p'-involve Qd(p) for any odd prime p. This follows from a more general theorem which allows us to construct a finite G-CW-complex by gluing together a given G-invariant family of representations defined on the Sylow subgroups of G.Comment: 16 page

Quotients of S2×S2S^2\times{S^2}

We consider closed topological 4-manifolds $M$ with universal cover ${S^2\times{S^2}}$ and Euler characteristic $\chi(M) = 1$. All such manifolds with $\pi=\pi_1(M)\cong {\mathbb Z}/4$ are homotopy equivalent. In this case, we show that there are four homeomorphism types, and propose a candidate for a smooth example which is not homeomorphic to the geometric quotient. If $\pi\cong {\mathbb Z}/2 \times {\mathbb Z}/2$, we show that there are three homotopy types (and between 6 and 24 homeomorphism types).Comment: 18 page

Cyclic branched coverings of Brieskorn spheres bounding acyclic 4-manifolds

We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic $4$-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well known invariants of homology spheres that obstruct acyclic bounding 4-manifolds, and then use a method based on equivariant Yang-Mills moduli spaces to rule out extensions of the actions

Group actions on spheres with rank one prime power isotropy

We show that a rank two finite group G admits a finite G-CWcomplex X Sn with rank one prime power isotropy if and only if G does not p-involve Qd(p) for any odd prime p. This follows from a more general theorem which allows us to construct a finite G-CW-complex by gluing together a given G-invariant family of representations defined on the Sylow subgroups of G. © by International Press of Boston, Inc. All rights reserved