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Title:
Turaev's loop operations on surfaces and their formal descriptions
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Abstract:
Turaev introduced in 1978 two operations on the fundamental group of a surface with boundary. The first operation measures the "homotopy" intersection of two loops on a surface; nowadays, it is known to control the Atiyah-Bott Poisson structures on representation spaces of surface groups, and to have generalizations for higher-dimensional manifolds. As for the second operation, it measures the "homotopy" self-intersection of a single loop, and it appears to be more mysterious than the first one. In this talk, we will survey these loop operations before addressing the problem of their "formal" descriptions. We will see that a "formality" isomorphism for the self-intersection operation of a punctured disk arises from any Drinfeld associator. The proof is based on some 3-dimensional formulas for Turaev's loop operations.
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