Deletion and contraction in configuration spaces of graphs

TL;DR

This paper develops functorial maps between configuration spaces of graphs induced by edge contraction and half-edge deletion, enabling algebraic and topological analysis of graph configuration spaces and their invariants.

Contribution

It introduces space-level maps for configuration spaces under graph minors, providing a framework for functoriality and inductive calculations in generalized (co)homology theories.

Findings

Long exact sequence for half-edge deletion in generalized cohomology

Functorial maps induced by graph minors at the homotopy level

Generalized homology of unordered configuration spaces is finitely generated

Abstract

The aim of this article is to provide space level maps between configuration spaces of graphs that are predicted by algebraic manipulations of cellular chains. More explicitly, we consider edge contraction and half-edge deletion, and identify the homotopy cofibers in terms of configuration spaces of simpler graphs. The construction's main benefit lies in making the operations functorial - in particular, graph minors give rise to compatible maps at the level of fundamental groups as well as generalized (co)homology theories. As applications we provide a long exact sequence for half-edge deletion in any generalized cohomology theory, compatible with cohomology operations such as the Steenrod and Adams operations, allowing for inductive calculations in this general context. We also show that the generalized homology of unordered configuration spaces is finitely generated as a representation of the opposite graph minor category.