The Euler characteristic of configuration spaces

TL;DR

This paper introduces a generating function for the compactly supported Euler characteristic of configuration spaces on stratified spaces, generalizing to equivariant cases and providing new computational tools.

Contribution

It presents a novel generating function formula for the Euler characteristic of configuration spaces with constructible sheaves, extending existing results to equivariant Euler characteristics.

Findings

Derived a generating function for $_c(F(X, n), K^{oxtimes n})$

Special case yields a generating function for $(F(X, n))$

Outlined how to compute equivariant Euler characteristics from the results

Abstract

In this short note we present a generating function computing the compactly supported Euler characteristic $\chi_c(F(X, n), K^{\boxtimes n})$ of the configuration spaces on a topologically stratified space $X$, with $K$ a constructible complex of sheaves on $X$, and we obtain as a special case a generating function for the Euler characteristic $\chi(F(X, n))$. We also recall how to use existing results to turn our computation of the Euler characteristic into a computation of the equivariant Euler characteristic.