On the Moduli Space of a Spherical Polygonal Linkage

Canadian mathematical bulletin, Tome 42 (1999) no. 3, pp. 307-320

Résumé

We give a “wall-crossing” formula for computing the topology of the moduli space of a closed $n$ -gon linkage on ${{\mathbb{S}}^{2}}$ . We do this by determining the Morse theory of the function ${{\rho }_{n}}$ on the moduli space of $n$ -gon linkages which is given by the length of the last side—the length of the last side is allowed to vary, the first $\left( n\,-\,1 \right)$ side-lengths are fixed. We obtain a Morse function on the $\left( n\,-\,2 \right)$ -torus with level sets moduli spaces of $n$ -gon linkages. The critical points of ${{\rho }_{n}}$ are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of ${{\rho }_{n}}$ at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.

Détail

DOI : 10.4153/CMB-1999-037-x

Mots-clés : 14D20, 14P05