Homology, Homotopy and Applications

Volume 17 (2015)

Number 2

On the middle dimensional homology classes of equilateral polygon spaces

Pages: 1 – 12

DOI: https://dx.doi.org/10.4310/HHA.2015.v17.n2.a1


Yasuhiko Kamiyama (Department of Mathematics, University of the Ryukyus, Nishihara-Cho, Okinawa, Japan)


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 where $n$ 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 $H_{m-1}(M_n;{\mathbb{Z}})$.


polygon space, involution, homology class

2010 Mathematics Subject Classification

55R80, 58D29

Published 3 December 2015