Tuesday, July 06, 2010

Smith theory made coarse

"Smith theory" is the name given to the investigation (starting with the work of P.A. Smith in the late 1930s) of the homological properties of the fixed-sets for finite groups (especially $p$-groups) acting on spheres. See Dwyer, William G., and Clarence W. Wilkerson. “Smith Theory Revisited.” The Annals of Mathematics 127, no. 1. Second Series (January 1988): 191-198.

Ian Hambleton and his student Lucian Savin have just posted an article about a coarse-geometric counterpart of Smith Theory, Hambleton, Ian, and Lucian Savin. “Coarse Geometry and P. A. Smith Theory.” 1007.0495 (July 3, 2010). http://arxiv.org/abs/1007.0495.

The notion of fixed set is replaced by a "coarse fixed set", which is the coarse structure (if it stabilizes) of the sequence of "approximate fixed sets" $\{x: d(x,f(x))\le n\}$ for $n=1,2,\ldots$.

This notion is well behaved for finite group actions.

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