Homology and Euler characteristic of generalized anchored configuration spaces of graphs

TL;DR

This paper studies the topology of generalized anchored configuration spaces on graphs, providing formulas for Euler characteristics and homology groups, especially for circle graphs, advancing understanding of configuration spaces in graph topology.

Contribution

It introduces a non-alternating formula for Euler characteristics of these spaces and fully determines their homology groups for circle graphs.

Findings

Derived a non-alternating Euler characteristic formula for connected non-tree graphs.

Determined the homology groups of configuration spaces on circle graphs.

Extended understanding of topological invariants of configuration spaces on graphs.

Abstract

In this paper we consider the generalized anchored configuration spaces on $n$ labeled points on a~graph. These are the spaces of all configurations of $n$ points on a~fixed graph $G$, subject to the condition that at least $q$ vertices in some pre-determined set $K$ of vertices of $G$ are included in each configuration. We give a non-alternating formula for the Euler characteristic of such spaces for arbitrary connected graphs, which are not trees. Furthermore, we completely determine the homology groups of the generalized anchored configuration spaces of $n$ points on a circle graph.