Volume 13, issue 5 (2013)

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Obstructions to Lagrangian cobordisms between Legendrians via generating families

Joshua M Sabloff and Lisa Traynor

Algebraic & Geometric Topology 13 (2013) 2733–2797

The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1–jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yields structural information about invariants of Legendrian submanifolds. For example, the obstructions devised in this paper show that there is no generating family compatible Lagrangian cobordism between the Chekanov–Eliashberg Legendrian m(52) knots. Further, the generating family cohomology groups of a Legendrian submanifold restrict the topology of a Lagrangian filling. Structurally, the generating family cohomology of a Legendrian submanifold satisfies a type of Alexander duality that, when the Legendrian is null-cobordant, can be seen as Poincaré duality of the associated Lagrangian filling. This duality implies the Arnold Conjecture for Legendrian submanifolds with linear-at-infinity generating families. These results are obtained by developing a generating family version of wrapped Floer cohomology and establishing long exact sequences that arise from viewing the spaces underlying these cohomology groups as mapping cones.

Lagrangian cobordism, Legendrian, generating family, duality
Mathematical Subject Classification 2010
Primary: 53D12, 57R17
Secondary: 57Q60
Received: 1 September 2012
Revised: 26 March 2013
Accepted: 27 March 2013
Published: 15 July 2013
Joshua M Sabloff
Department of Mathematics and Statistics
Haverford College
370 Lancaster Ave
Haverford, PA 19041
Lisa Traynor
Department of Mathematics
Bryn Mawr College
Bryn Mawr, PA 19010