The Euler characteristic of the regular spherical polygon spaces (Q2284270)

\textit{M. Kapovich} and \textit{J. Millson} in [J. Differ. Geom. 42, No. 1, 133--164 (1995; Zbl 0847.51026)] studied spherical polygon spaces, by means of a certain configuration space. They derived a formula for the signature of \(D^2\rho_2|_{T}\) for a certain real valued function \(\rho_n\) defined on this configuration space. To study the subfamily of regular spherical polygons the work under review considers the following configuration space: Given \(0<a<\pi\) define \[ M_n(a)=\{u=(u_1,\dots,u_n)\in (S^2)^n \ | \ d(u_i, u_{i+1})=a \text{ for } i=1,\dots,n-1, \text{ and } d(u_1,u_n)=a\}, \] which is called the regular spherical polygon space. The goal of the present work is to compute the Euler characteristic \(\chi(M_n(a)\)). In order to do that the author uses the following results which are the two main results of the work: Theorem 1.3. An element \((P,a)\) is a critical point of \(\mu\) if and only if \(P\) is degenerated and Theorem 1.4. Let \(P\in M_n(a)\) be a degenerate polygon. Then the following hold: (i) When \(w(P)>0\), the signature of \(D^2\mu|_{(P,a)}\) is given by \[ (f(P)-2w(P)-1, b(P)+2w(P)-1). \] (ii) When \(w(P)<0\), the signature of \(D^2\mu|_{(P,a)}\) is given by \[ (b(P)+2w(P)-1, f(P)-2w(P)-1). \] Here \(\mu:M_n(a)\to R\) is a certain real valued function. At the end some explicit formulas for the Euler characteristic are derived for the cases \(\chi(M_n(\pi/2))\), \(\chi(M_n(a))\) where \(n\) is odd and \(0<a<2\pi/n\).