Cohomology of Polychromatic Configuration Spaces of Euclidean Space

TL;DR

This paper extends the understanding of the homology and cohomology of configuration spaces by analyzing polychromatic and bicolored variants, allowing for diverse behaviors among points in Euclidean space.

Contribution

It introduces calculations for homology and cohomology of decreasing polychromatic and bicolored configuration spaces, generalizing previous work on non-k-overlapping discs.

Findings

Calculated homology and cohomology for polychromatic configuration spaces

Extended results to bicolored configuration spaces

Demonstrated varied point behaviors in configuration space topology

Abstract

Recently, the homology and cohomology of non-k-overlapping discs, or, equivalently, no k-equal subspaces of Euclidean space, were calculated by Dobrinskaya and Turchin. We calculate the homology and cohomology of two classes of more general spaces: decreasing polychromatic configuration spaces and bicolored configuration spaces. Instead of all points behaving similarly, we allow for varying behavior between points.