On the middle dimensional homology classes of equilateral polygon spaces

Abstract: Let

M_{n}

be the configuration space of equilateral polygonal linkages with

n

vertices in the Euclidean plane

{m a t h b b R}^{2}

. We consider the case that

n

is odd and set

n = 2 m + 1

. In spite of the long history of research, the homology classes in

H_{m - 1} (M_{n}; m a t h b b Z)

are mysterious and not well-understood. Let

a u c o l o n M_{n} o M_{n}

be the involution induced by complex conjugation. In this paper, we determine the representation matrix of the homomorphism

a u_{a} s t c o l o n H_{m - 1} (M_{n}; m a t h b b Z) o H_{m - 1} (M_{n}; m a t h b b Z)

with respect to a basis of

H_{m - 1} (M_{n}; m a t h b b Z)

.