A Stable Infinity-Category of Lagrangian Cobordisms

Hiro Tanaka, Northwestern University / QGM

2012.11.07 | Christine Dilling

Date Wed 16 Nov
Time 16:15 17:15
Location Aud. D3

Abstract

In this talk Hiro Tanaka will discuss a joint project with David Nadler. They construct a category Lag(M), which one can associate to any exact symplectic manifold M. They conjecture that this category is equivalent to a Fukaya category of M, and the result he'll discuss in this talk is that Lag(M) is a stable infinity-category in the sense of Lurie. Some consequences are that Lag(M) is enriched over spectra, and that the homotopy category of Lag(M) is triangulated. They also show that the shift functor, on objects, is equivalent to that of other Fukaya categories in which Lagrangians are graded. Roughly speaking, the objects of Lag(M) are exact Lagrangian submanifolds, and morphisms are cobordisms between them which are Lagrangians in M x T*R. The cobordisms must also be non-characteristic with respect to a Lagrangian skeleton of M. (There are also variations of Lag(M) given by varying the choice of Lagrangian skeleton Lambda.) If time allows, he may also discuss future work, some of which involves computing a Thom spectrum related to (non-compact) Lagrangian cobordisms--this spectrum acts as the `universal coefficients' of Lag(M).

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