多边形空间的相交数
Intersection Numbers of Polygon Spaces
摘要：
研究了多边形空间$\M(α_1,...,α_m)$在$\R^3$中的交环。找到了与该环的生成元对偶的同调圈，并证明了它们的相交数在$m$（步数）中的一个递归关系。这一结果是类似的递归关系出现在工作的威滕和Kontsevich模空间的穿孔曲线和工作的魏茨曼模空间的平坦连接的两个流形的属$g$与$m$标记点。基于这个递推公式，我们得到了计算多边形空间的相交数的一个显式表达式，并在几个例子中加以应用。除其他外，我们研究的特殊情况下，等边多边形空间（其中所有的$α_i$是相同的），并比较我们的结果与这些特定的空间，已确定由神山和手冢的表达式。最后，我们将我们的相交数的显式公式与Yoshida的平坦联络的模空间的相交配对的生成函数以及与Takakura和Konno通过不同技巧得到的多边形空间的等价表达式联系起来.
英文摘要：
We study the intersection ring of the space $\M(α_1,...,α_m)$ 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 $α_i$ 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.
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