# On the Topology of Symmetric Semialgebraic Sets

This work strengthens and extends an algorithm for computing Betti numbers of symmetric semialgebraic sets developed by Basu and Riener in, *Vandermonde Varieties, Mirrored Spaces, and the Cohomology of Symmetric Semi-Algebraic Sets*. We first adapt a construction of Gabrielov and Vorobjov in, *Approximation of Definable Sets by Compact Families, and Upper Bounds on Homotopy and Homology,* for replacing arbitrary definable sets by compact ones to the symmetric case. The original construction provided maps from the homotopy and homology groups of the replacement set to those of the original; we show that for sets symmetric relative to the action of some finite reflection group *G*, we may construct these maps to be equivariant. This modification to the construction for compact replacement allows us to extend Basu and Riener's theorem on which submodules appear in the isotypic decomposition of each cohomology space to sets not necessarily closed and bounded. Furthermore, by utilizing this equivariant compact approximation, we may obtain a precise description of the aforementioned decomposition of each cohomology space, and not merely the final dimension of the space, from Basu and Riener's algorithm.

Though our equivariant compact replacement holds for *G* any finite reflection group, Basu and Riener's results only consider the case of the action the of symmetric group, sometimes termed type *A*. As a first step towards generalizing Basu and Riener's work, we examine the next major class of symmetry: the action of the group of signed permutations (known as type *B*). We focus our attention on Vandermonde varieties, a key object in Basu and Riener's proofs. We show that the intersection of a type *B* Vandermonde variety with a fundamental region of type *B* symmetry is topologically regular. We also prove a result about the intersection of a type *B* Vandermonde variety with the walls of this fundamental region, leading to the elimination of factors in a different decomposition of the homology spaces.

## History

## Degree Type

- Doctor of Philosophy

## Department

- Mathematics

## Campus location

- West Lafayette