On the middle dimensional homology classes of equilateral polygon spaces

TL;DR

This paper investigates the middle-dimensional homology classes of equilateral polygon spaces with an odd number of vertices, specifically analyzing the involution induced by complex conjugation on these classes.

Contribution

It determines the representation matrix of the involution on the middle-dimensional homology classes of equilateral polygon spaces for odd n.

Findings

Explicit matrix representation of the involution on homology classes

Enhanced understanding of the structure of homology in polygon spaces

Progress towards resolving longstanding questions about these homology classes

Abstract

Let $M_n$ be the configuration space of equilateral polygonal linkages with $n$ vertices in the Euclidean plane ${\mathbb R}^2$. We consider the case that $n$ is odd and set $n=2m+1$. In spite of the long history of research, the homology classes in $H_{m-1}(M_n;{\mathbb Z})$ are mysterious and not well-understood. Let $\tau\colon M_n \to M_n$ be the involution induced by complex conjugation. In this paper, we determine the representation matrix of the homomorphism $\tau_\ast\colon H_{m-1}(M_n;{\mathbb Z}) \to H_{m-1}(M_n;{\mathbb Z})$ with respect to a basis of $H_{m-1}(M_n;{\mathbb Z})$.