The Euler characteristic of the regular spherical polygon spaces

Let $a$ be a real number satisfying $0 \lt a \lt \pi$. 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 $\mu : X_n \to \mathbb{R}$ such that $\mu^{-1} (a) = M_n (a)$. In fact, the function μ is different from the Kapovich–Millson Morse function. We determine the index of each critical point of $\mu$. Since a level set is obtained by successive Morse surgeries, we can determine $\chi (M_n (a))$.

Keywords

spherical polygon space, Morse function, Euler characteristic

2010 Mathematics Subject Classification

58D29, 58E05

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Received 14 March 2018

Received revised 6 April 2019

Accepted 21 April 2019

Published 18 September 2019