Wrapped sutured Legendrian homology and unit conormal of local 2-braids
C\^ome Dattin

TL;DR
This paper develops new Legendrian homology invariants for sutured Legendrians with boundary, extending Floer theory techniques to define and relate cylindrical, wrapped, and Chekanov-Eliashberg invariants, with applications to classifying local 2-braids.
Contribution
It introduces a sutured Legendrian framework with invariants derived from Floer theory, including an exact sequence and a wrapped Chekanov-Eliashberg algebra, for Legendrians with boundary.
Findings
Invariants are preserved under boundary-fixed Legendrian isotopy.
The sutured Legendrian invariants distinguish certain manifold features.
Isotopic conormals imply braid equivalence in local 2-braids.
Abstract
We extend the sutured framework to the case of Legendrians with boundary. Using ideas from Lagrangian Floer theory, we define the cylindrical and the wrapped sutured Legendrian homologies of a pair of sutured Legendrians. They fit together into an exact sequence, and the exact triangle is invariant along an Legendrian isotopy fixed at the boundary. For a single Legendrian, we also define a wrapped version of its Chekanov-Eliashberg dga. Our main example of sutured Legendrian is obtained via the unit conormal construction : a submanifold , such that , induces a sutured Legendrian , thus we get smooth invariants of manifolds with boundary. As a simple application, we show that if the conormals of two local 2-braids are isotopic (as Legendrians with fixed boundary), then the braids are equivalent.
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometric Analysis and Curvature Flows · Homotopy and Cohomology in Algebraic Topology
