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Homology and Euler characteristic of generalized anchored configuration spaces of graphs

Dmitry N. Kozlov

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.

Homology and Euler characteristic of generalized anchored configuration spaces of graphs

Abstract

In this paper we consider the generalized anchored configuration spaces on labeled points on a~graph. These are the spaces of all configurations of points on a~fixed graph , subject to the condition that at least vertices in some pre-determined set of vertices of 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 points on a circle graph.
Paper Structure (8 sections, 8 theorems, 22 equations)

This paper contains 8 sections, 8 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. The Euler characteristic of the generalized anchored configuration spaces
  3. The chain complexes for the generalized anchored configuration spaces on circle graphs
  4. Calculation of the homology groups of $\mathcal{C}^{P,q}$
  5. The case $q=0$
  6. Structure of the relative chain complexes
  7. The case $P\neq{\mathbb Z}_k$
  8. The case $P={\mathbb Z}_k$

Key Result

Theorem 2.1

Let $G$ be an arbitrary connected graph, whose set of vertices is $V$, and whose set of edges is $E$. Let $K$ be an arbitrary non-empty subset of $V$, and let $q$ be a positive integer, such that $q\leq |K|$. Finally, let $n$ be a natural number, such that $n\geq q$. Assume the graph $G$ is not a tr where $k:=|K|$ and $\varepsilon:=|E|-|V|$.

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Corollary 2.2
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Proposition 4.1
  • Lemma 4.2
  • Theorem 4.3
  • ...and 3 more