Algebraic & Geometric Topology

Obstructions to Lagrangian cobordisms between Legendrians via generating families

Joshua M Sabloff and Lisa Traynor

Full-text: Open access

Abstract

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.

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 5 (2013), 2733-2797.

Dates
Received: 1 September 2012
Revised: 26 March 2013
Accepted: 27 March 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715689

Digital Object Identifier
doi:10.2140/agt.2013.13.2733

Mathematical Reviews number (MathSciNet)
MR3116302

Zentralblatt MATH identifier
1270.53096

Subjects
Primary: 53D12: Lagrangian submanifolds; Maslov index 57R17: Symplectic and contact topology
Secondary: 57Q60: Cobordism and concordance

Keywords
Lagrangian cobordism Legendrian generating family duality

Citation

Sabloff, Joshua M; Traynor, Lisa. Obstructions to Lagrangian cobordisms between Legendrians via generating families. Algebr. Geom. Topol. 13 (2013), no. 5, 2733--2797. doi:10.2140/agt.2013.13.2733. https://projecteuclid.org/euclid.agt/1513715689


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