Homology of moduli spaces of linkages in high-dimensional Euclidean space

TL;DR

This paper investigates the topological properties of moduli spaces of closed linkages in high-dimensional Euclidean spaces, using Morse theory to analyze their homology, especially focusing on the case when the dimension is five.

Contribution

It applies equivariant Morse theory to compute homology of linkage moduli spaces and explicitly calculates the Poincare polynomial for five-dimensional cases.

Findings

Homology groups characterized for odd dimensions.

Poincare polynomial explicitly computed for d=5.

Topological insights into linkage spaces derived from combinatorial data.

Abstract

We study the topology of moduli spaces of closed linkages in \R^d depending on a length vector \ell\in \R^n. In particular, we use equivariant Morse theory to obtain information on the homology groups of these spaces, which works best for odd d. In the case d=5 we calculate the Poincare polynomial in terms of combinatorial information encoded in the length vector.