Abstract. We give a general framework for studying G-CW complexes via the orbit category. As an application we show that the symmetric group G = S 5 admits a finite G-CW complex X homotopy equivalent to a sphere, with cyclic isotropy subgroups.
We show that there is an exact sequence of biset functors over p-groupswhere C b is the biset functor for the group of Borel-Smith functions, B * is the dual of the Burnside ring functor, D Ω is the functor for the subgroup of the Dade group generated by relative syzygies, and the natural transformation Ψ is the transformation recently introduced by the first author in [S. Bouc, A remark on the Dade group and the Burnside group, J. Algebra 279 (2004) 180-190]. We also show that the kernel of mod 2 reduction of Ψ is naturally equivalent to the functor B × of units of the Burnside ring and obtain exact sequences involving the torsion part of D Ω , mod 2 reduction of C b , and B × .
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.
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