Room P3.10, Mathematics Building

Alessia Mandini, Instituto Superior Tecnico
The Cobordism Class and the Symplectic Volume of the Moduli Space of Polygons in $\mathbb{R}^3$

For any vector $r=(r_1,\ldots,r_n)$, let $M_r$ denote the moduli space (under rigid motions) of polygons in $\mathbb{R}^3$ with $n$ sides whose lengths are $r_1,\ldots,r_n$. We give an explicit characterization of the oriented $S^1$-cobordism class of $M_r$ which depends uniquely on the length vector $r$. The main ingredients to prove our result are the bending action (Kapovich-Millson) and results presented by Ginzburg, Guillemin and Karshon that, under suitable hypothesis, link the $S^1$-cobordism class of an even-dimensional manifold $M$ to data associated to the connected components of the fixed point set $(M)^{S^1}$. Also we prove a formula expressing the volume of $M_r$ as a piecewise polynomial function in the $r_i$ (in accordance with the Duitermaat-Heckman Theorem). The main tool to prove this result are localization theorems in equivariant cohomology (Martin, Jeffrey-Kirwan, Guillemin-Kalkman) together with an equivariant integration formula for symplectic quotients by non-abelian groups (Martin).