Equivariance in Approximation by Compact Sets

Abstract

We adapt a construction of Gabrielov and Vorobjov for use in the symmetric case. Gabrielov and Vorobjov had developed a means by which one may replace an arbitrary set $S$ definable in some o-minimal expansion of $\mathbb{R}$ with a compact set $T$. $T$ is constructed in such a way that for a given $m>0$ we have epimorphisms from the first $m$ homotopy and homology groups of $T$ to those of $S$. If $S$ is defined by a boolean combination of statements $h(x)=0$ and $h(x)>0$ for various $h$ in some finite collection of definable continuous functions, one may choose $T$ so that these maps are isomorphisms for $0\leq k\leq m-1$. In this case, $T$ is also defined by functions closely related to those defining $S$. In this paper we study sets $S$ symmetric under the action of some finite reflection group $G$. One may see that in the original construction, if $S$ is defined by functions symmetric relative to the action of $G$, then $T$ will be as well. We show that there is an equivariant map $T\rightarrow S$ inducing the aforementioned epimorphisms and isomorphisms of homotopy and homology groups. We use this result to strengthen theorems of Basu and Riener concerning the multiplicities of Specht modules in the isotypic decomposition of the cohomology spaces of sets defined by polynomials symmetric relative to $\mathfrak{S}_n$.