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# Homology, Homotopy and Applications

## Volume 17 (2015)

### Number 2

### On the middle dimensional homology classes of equilateral polygon spaces

Pages: 1 – 12

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

#### Author

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

#### Keywords

polygon space, involution, homology class

#### 2010 Mathematics Subject Classification

55R80, 58D29