标题： 带框架的三维流形关于三价图（可能带有自环）的积分不变量 显示英文标题

标题： Integral invariants for framed 3-manifolds associated to trivalent graphs possibly with self-loops

摘要： Bott--Cattaneo理论通过无自环的图循环定义了带有可约正交局部系的框架有理同调3-球体的积分不变量。 他们的不变量的2环项与θ图相关。 他们的定义需要一个上同调条件。 Cattaneo--Shimizu去除了这个上同调条件，并给出了一个与θ图和哑铃图的线性组合相关的2环不变量，即带有自环的2环三价图。 在本文中，我们关注的是由半单李群的伴随表示与闭3-流形基本群的表示组成的可约局部系，并且我们证明通过最终的上同调构造，与哑铃图相关的积分消失。 基于这个想法，我们构建了一个图复形和循环的理论，使得高环不变量可以通过两种不同但等价的方法来定义：如Bott--Cattaneo理论中的无自环的图循环，以及扩展Cattaneo--Shimizu的2环不变量的带有自环的循环。 作为结果，我们证明了Chern--Simons微扰理论的生成级数在我们的设定下产生了框架3-流形的拓扑不变量，该不变量仅能用无自环的三价图来表示。 显示英文摘要

摘要： Bott--Cattaneo's theory defines the integral invariants for a framed rational homology 3-sphere equipped with an acyclic orthogonal local system, in terms of graph cocycles without self-loops. The 2-loop term of their invariants is associated with the theta graph. Their definition requires a cohomological condition. Cattaneo--Shimizu removed this cohomological condition and gave a 2-loop invariant associated with a linear combination of the theta graph and the dumbbell graph, the 2-loop trivalent graph with self-loops. In this paper, we are concerned with an acyclic local system given by the adjoint representation of a semi-simple Lie group composed with a representation of the fundamental group of a closed 3-manifold, and we show that through a cohomological construction eventually the integral associated with the dumbbell graph vanishes. Based on this idea, we construct a theory of graph complexes and cocycles, so that higher-loop invariants can be defined by two different but equivalent methods: the graph cocycles without self-loops as in Bott--Cattaneo's theory, and the ones with self-loops that extend Cattaneo--Shimizu's 2-loop invariants. As a consequence, we prove that the generating series of Chern--Simons perturbation theory gives rise to topological invariants for framed 3-manifolds in our setting, which admits a formula in terms of only trivalent graphs without self-loops.