We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen’s iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a commutative differential graded algebra suitable for integration. We show that this integration is compatible with Bott–Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids.

中文翻译：

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辫形空间的图复杂度，形式性和配置空间积分

我们使用配置空间的合理形式和钢筋构造来研究尺寸为4或更大的编织空间的同调性。我们为辫子提供了一个图复杂体，并为辫子空间上的de Rham协链提供了一个准同构。准同构由配置空间积分和Chen的迭代积分给出。这将Kohno以及Cohen和Gitler的关于辫子空间的同调性的结果扩展到了适合积分的可交换微分渐变代数。我们通过两个图复合体之间的映射，证明了这种集成与长链接的Bott-Taubes配置空间积分兼容。作为必然结果，我们从长链接空间到辫子空间得到了同调的推论。