Intersection Numbers of Polygon Spaces

TL;DR

This paper investigates the intersection ring of polygon spaces in three dimensions, deriving a recursion relation for intersection numbers, providing explicit formulas, and connecting these results to related moduli space theories.

Contribution

It introduces a recursion relation for intersection numbers in polygon spaces and provides explicit formulas, linking these to existing moduli space frameworks.

Findings

Derived a recursion relation for intersection numbers.

Provided explicit formulas for intersection computations.

Connected polygon space intersection numbers to moduli space generating functions.

Abstract

We study the intersection ring of the space $\M(\alpha_1,...,\alpha_m)$ of polygons in $\R^3$. We find homology cycles dual to generators of this ring and prove a recursion relation in $m$ (the number of steps) for their intersection numbers. This result is analog of the recursion relation appearing in the work of Witten and Kontsevich on moduli spaces of punctured curves and on the work of Weitsman on moduli spaces of flat connections on two-manifolds of genus $g$ with $m$ marked points. Based on this recursion formula we obtain an explicit expression for the computation of the intersection numbers of polygon spaces and use it in several examples. Among others, we study the special case of equilateral polygon spaces (where all the $\alpha_i$ are the same) and compare our results with the expressions for these particular spaces that have been determined by Kamiyama and Tezuka. Finally, we relate our explicit formula for the intersection numbers with the generating function for intersection pairings of the moduli space of flat connections of Yoshida, as well as with equivalent expressions for polygon spaces obtained by Takakura and Konno through different techniques.