On the middle dimensional homology classes of equilateral polygon spaces

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 $\tau \text{\hspace{0.17em}⁣}:M_n\to M_n$ be the involution induced by complex conjugation. In this paper, we determine the representation matrix of the homomorphism ${\tau }_{\ast }\text{\hspace{0.17em}⁣}:H_{m-1}(M_n;\mathbb{Z})\to H_{m-1}(M_n;\mathbb{Z})$ with respect to a basis of $H_{m-1}(M_n;\mathbb{Z})$.