Configuration space in a product

TL;DR

This paper studies the topology of configuration spaces on product spaces, providing homotopy decompositions, stabilization results, and explicit homology computations, with applications to complex tori and algorithms for simplicial complexes.

Contribution

It introduces a homotopy decomposition for configuration spaces on product spaces and develops an algorithm for homology computation in these contexts.

Findings

Homotopy decomposition of Conf(G, X x Y) in terms of Conf(G, X) and Conf(G, Y)

Homology stabilization for Conf(X x C^p) as p increases

Explicit homology calculations for configurations in tori

Abstract

Given a finite graph G and a topological space Z, the graphical configuration space Conf(G, Z) is the space of functions V(G) -> Z so that adjacent vertices map to distinct points. We provide a homotopy decomposition of Conf(G, X x Y) in terms of the graphical configuration spaces in X and Y individually. By way of application, we prove a stabilization result for homology of configuration space in X x C^p as p goes to infinity. We also compute the homology of Conf(K_3,T)/T, the space of ordered triples of distinct points in a torus T of rank r, where configurations are considered up to translation. In Section 2, we give an algorithm for computing homology of configuration space in a product of simplicial complexes. The method is applied to products of some sans-serif capital letters in Example 2.12.