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Lagrangians, SO(3)-instantons and the Atiyah-Floer Conjecture

Banff International Research Station for Mathematical Innovation and Discovery

A useful tool to study a 3-manifold is the space of representations of its fundamental group into a Lie group. Any 3-manifold can be decomposed as the union of two handlebodies. Thus representations of the 3-manifold group into a Lie group can be obtained by intersecting representation varieties of the two handlebodies. Casson utilized this observation to define his celebrated invariant. Later Taubes introduced an alternative approach to define Casson invariant using more geometric objects. By building on Taubes' work, Floer refined Casson invariant into a 3-manifold invariant which is known as instanton Floer homology. The Atiyah-Floer conjecture states that Casson's original approach can be also used to define a graded vector space and the resulting invariant of 3-manifolds is isomorphic to instanton Floer homology. In this talk, I will discuss a variation of the Atiyah-Floer conjecture, which states that framed Floer homology (defined by Kronheimer and Mrowka) is isomorphic to symplectic framed Floer homology (defined by Wehrheim and Woodward). I will also discuss how techniques from symplectic topology could be useful to study framed Floer homology. This talk is based on a joint work with Kenji Fukaya and Maksim Lipyanskyi.

Mathematics; Manifolds and cell complexes; Global analysis, analysis on manifolds

video/mp4

Author affiliation: Washington University

2020-12-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/76770

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/