The Euler characteristic of the regular spherical polygon spaces

Let $a$ be a real number satisfying $0<a<\pi$. We denote by $M_n(a)$ the configuration space of regular spherical $n$-gons with side-lengths $a$. The purpose of this paper is to determine $\chi (M_n(a))$ for all $a$ and odd $n$. To do so, we construct a manifold $X_n$ and a function $\mu: X_n \to \mathbf{R}$ such that $\mu^{-1}(a)=M_n(a)$. In fact, the function $\mu$ is different from the well-known "wall-crossing" function. We determine the index of each critical point of $\mu$. Since a level set is obtained by successive Morse surgeries, we can determine $\chi (M_n(a))$.