seminar: Surgery formulae for 3-manifold invariants defined via configuration spaces
speaker: Christine Lescop (Institut Fourier, CNRS, Grenoble France)abstract: The Casson invariant $\lambda$ is a topological invariant of integral homology 3-spheres (that are 3-manifolds with the same integral homology as $S3$). Casson defined it by introducing a clever way of counting the SU(2)-representations of the fundamental groups of these manifolds, in 1984. In 1999, Greg Kuperberg and Dylan Thurston showed how to express the Casson invariant as a configuration space integral.
This Kuperberg-Thurston result implies that $6 \lambda(M)$ is the algebraic intersection of three codimension 2 manifolds in the 6-dimensional space of two-point configurations of M, for an integral homology sphere M. I shall explain this result and its extension to the Walker generalisation of the Casson invariant to rational homology spheres.
More precisely, I shall give a self-contained presentation of the Casson-Walker invariant, including a direct proof of the Walker surgery formula, in the setting of configuration spaces.
In this setting, the Casson-Walker invariant is "the" degree one invariant among the invariants of rational homology spheres that were defined by Maxim Kontsevich, Greg Kuperberg and Dylan Thurston, using configuration space integrals. The Casson-Walker surgery formula together with its mentioned direct proof generalises to surgery formulae for surgeries along $n$-component boundary links in rational homology spheres for all the degree $n$ invariants. I shall present these formulae, too.
timetable:
Thu 8 Sep, 10:30 - 11:30, Aula Dini
documents:
Surgery formulae for 3-manifold invariants defined via configuration spaces
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