Intersection homology of linkage spaces

TL;DR

This paper investigates the intersection homology of linkage moduli spaces in higher dimensions, providing tools to distinguish their topological types, especially when the space is a pseudomanifold rather than a manifold.

Contribution

It introduces a ring structure via intersection homology for linkage spaces in dimensions greater than three, extending the Walker conjecture to higher dimensions.

Findings

The ring distinguishes homeomorphism types for many length vectors.

The spaces are pseudomanifolds for d>3, not manifolds.

Extension of the Walker conjecture to high-dimensional linkage spaces.

Abstract

We consider the moduli spaces $\mathcal{M}_d(\ell)$ of a closed linkage with n links and prescribed lengths 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 type of the moduli spaces for a large class of length vectors in the case of d even. This result is a high-dimensional analogue of the Walker conjecture which was proven by Farber, Hausmann and the author.