胖图算法与Kontsevich复形的同调
Fatgraph Algorithms and the Homology of the Kontsevich Complex
摘要：
胖图是一种由与顶点关联的边的循环序所丰富的多重图。本文给出了以下算法：（1）生成具有给定亏格和边界圈数的所有胖图的集合;（2）计算任意给定胖图的自同构;（3）计算胖图复形的同调。该算法适合于有效的计算机实现。特别是，这使我们能够计算的有理同调的模空间的黎曼曲面的标记点。因此，我们用$(2g + n) \leq 6$计算$M_{g,n}$的贝蒂数，证实了已知的结果。
英文摘要：
Fatgraphs are multigraphs enriched with a cyclic order of the edges incident to a vertex. This paper presents algorithms to: (1) generate the set of all fatgraphs having a given genus and number of boundary cycles; (2) compute automorphisms of any given fatgraph; (3) compute the homology of the fatgraph complex. The algorithms are suitable for effective computer implementation. In particular, this allows us to compute the rational homology of the moduli space of Riemann surfaces with marked points. We thus compute the Betti numbers of $M_{g,n}$ with $(2g + n) \leq 6$, corroborating known results.
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