Intersection homology of linkage spaces in odd dimensional Euclidean space

TL;DR

This paper studies the intersection homology of linkage spaces in odd-dimensional Euclidean spaces, revealing differences in their topological invariants compared to even dimensions, and introduces a ring structure to distinguish their homeomorphism types.

Contribution

It extends the intersection homology framework to odd-dimensional linkage spaces, identifying a new generator in the associated ring related to an Euler class, and differentiates odd from even dimensional cases.

Findings

The intersection homology ring distinguishes homeomorphism types of linkage spaces.

Odd-dimensional linkage spaces have an additional generator in their intersection homology ring.

The difference between even and odd dimensions is characterized by an Euler class in the ring.

Abstract

We consider the moduli spaces $\mathcal{M}_d(\ell)$ of a closed linkage with $n$ links and prescribed lengths $\ell\in \mathbb{R}^n$ in $d$-dimensional Euclidean space. For $d>3$ these spaces are no longer manifolds generically, but they have the structure of a pseudomanifold. We use intersection homology to assign a ring to these spaces that can be used to distinguish the homeomorphism types of $\mathcal{M}_d(\ell)$ for a large class of length vectors. These rings behave rather differently depending on whether $d$ is even or odd, with the even case having been treated in an earlier paper. The main difference in the odd case comes from an extra generator in the ring which can be thought of as an Euler class of a startified bundle.