图的复形上的无穷小运算
Infinitesimal Operations on Complexes of Graphs
摘要：
在两个开创性的文件Kontsevich使用了建设所谓的_图同源_作为桥梁之间的某些无限维李代数和各种拓扑对象，包括模空间的曲线，该集团的外部自同构的自由群，和不变量的奇维流形。在本文中，我们表明，Kontsevich的图复合体，其中包括图复合体研究较早的Culler和Vogtmann和Penner，有丰富的代数结构。我们在图复形上定义了一个李括号和余括号，并证明了它们是Batalin-Vilkovisky代数，因此也是Gerstenhaber代数。我们还发现自然子复形上的括号和cobracket是兼容的李双代数。Kontsevich的图复形构造由Ginzburg和Kapranov推广到运算的上下文中，后来由Getzler-Kapranov和Markl推广。在[CoV]中，我们证明了Kontsevich的结果实际上扩展到一般的循环运算。对于某些运算，包括模空间和自由群的外自同构群的例子，具有李双代数结构的子复形与整个连通图复形是拟同构的。本文证明了当应用同调函子时，所有新的代数运算都正则为零，并期望由此得到的约束条件在研究映射类群的同调、有限型流形不变量和Out（F_n）的同调等问题中有一定的应用.
英文摘要：
In two seminal papers Kontsevich used a construction called_graph homology_ as a bridge between certain infinite dimensional Lie algebras and various topological objects, including moduli spaces of curves, the group of outer automorphisms of a free group, and invariants of odd dimensional manifolds. In this paper, we show that Kontsevich's graph complexes, which include graph complexes studied earlier by Culler and Vogtmann and by Penner, have a rich algebraic structure. We define a Lie bracket and cobracket on graph complexes, and in fact show that they are Batalin-Vilkovisky algebras, and therefore Gerstenhaber algebras. We also find natural subcomplexes on which the bracket and cobracket are compatible as a Lie bialgebra. Kontsevich's graph complex construction was generalized to the context of operads by Ginzburg and Kapranov, with later generalizations by Getzler-Kapranov and Markl. In [CoV], we show that Kontsevich's results in fact extend to general cyclic operads. For some operads, including the examples associated to moduli space and outer automorphism groups of free groups, the subcomplex on which we have a Lie bi-algebra structure is quasi-isomorphic to the entire connected graph complex. In the present paper we show that all of the new algebraic operations canonically vanish when the homology functor is applied, and we expect that the resulting constraints will be useful in studying the homology of the mapping class group, finite type manifold invariants and the homology of Out(F_n).
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