Configuration spaces of labeled points on a circle with two anchors

TL;DR

This paper computes the homology of configuration spaces of points on a circle with two fixed points, using discrete Morse theory to determine Betti numbers and provide explicit bases for homology and cohomology.

Contribution

It introduces a novel application of discrete Morse theory to explicitly compute and describe the homology of constrained configuration spaces on a circle.

Findings

Betti numbers of the configuration space are explicitly determined.

Explicit combinatorial bases for homology and cohomology are constructed.

Homology calculations are achieved through discrete Morse theory techniques.

Abstract

In this paper we calculate the homology of configuration spaces of $n$ points on a circle, subject to the condition that two pre-determined points are included in the configuration. We make use of discrete Morse theory both to determine the Betti numbers, as well as to provide an explicit combinatorial description of the bases both for homology and cohomology.