Borel-Moore Homology of 2-Particle Unordered Configuration Spaces of Graphs via Discrete Morse Theory

Abstract

We define a discrete Morse function on the open regular cell complex $C_2(\Gamma)$, arising from the $2$-particle unordered configuration space of a graph $\Gamma$. We compute the homology of the associated Morse complex. Using an isomorphism established by Knudson and Scoville between the Borel-Moore homology of $C_2(\Gamma)$ and the homology of the Morse complex, we conclude that the Borel-Moore homology groups of $C_2(\Gamma)$ are completely determined by the first Betti number of $\Gamma$.