Abstract.A representation G ⊂ U (n) of degree n has fixity equal to the smallest integer f such that the induced action of G on U (n)/U (n − f − 1) is free. Using bundle theory we show that if G admits a representation of fixity one, then it acts freely and smoothly on S 2n−1 × S 4n−5 . We use this to prove that a finite p-group (for p > 3) acts freely and smoothly on a product of two spheres if and only if it does not contain (Z/p) 3 as a subgroup.We use propagation methods from surgery theory to show that a representation of fixity f < n − 1 gives rise to a free action of G on a product of f + 1 spheres provided the order of G is relatively prime to (n − 1)! . We give an infinite collection of new examples of finite p-groups of rank r which act freely on a product of r spheres, hence verifying a strong form of a well-known conjecture for these groups. In addition we show that groups of fixity two act freely on a finite complex with the homotopy type of a product of three spheres. A number of examples are explicitly described.
Mathematics Subject Classification (2000). 57S25.
Abstract. We show that every rank two p-group acts freely and smoothly on a product of two spheres. This follows from a more general construction: given a smooth action of a finite group G on a manifold M , we construct a smooth free action on M ×S n1 ×· · ·×S n k when the set of isotropy subgroups of the G-action on M can be associated to a fusion system satisfying certain properties. Another consequence of this construction is that if G is an (almost) extra-special p-group of rank r, then it acts freely and smoothly on a product of r spheres.
Abstract. For some small values of f , we prove that if G is a group having a complex (real) representation with fixity f , then it acts freely and smoothly on a product of f + 1 spheres with trivial action on homology.
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