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Chekanov-Eliashberg Dg-Algebras and Partially Wrapped Floer Cohomology

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Chekanov-Eliashberg Dg-Algebras and Partially Wrapped Floer Cohomology UPPSALA DISSERTATIONS IN MATHEMATICS 120 Chekanov-Eliashberg dg-algebras and partially wrapped Floer cohomology Johan Asplund Department of Mathematics Uppsala University UPPSALA 2021 Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 4 June 2021 at 14:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Dr, HDR Baptiste Chantraine (Département de Mathématiques, Universite de Nantes, France). Abstract Asplund, J. 2021. Chekanov-Eliashberg dg-algebras and partially wrapped Floer cohomology. Uppsala Dissertations in Mathematics 120. 42 pp. Uppsala: Department of Mathematics. ISBN 978-91-506-2872-2. This thesis consists of an introduction and two research papers in the fields of symplectic and contact geometry. The focus of the thesis is on Floer theory and symplectic field theory. In Paper I we show that the partially wrapped Floer cohomology of a cotangent fiber stopped by the unit conormal of a submanifold, is equivalent to chains of based loops on the complement of the submanifold in the base. For codimension two knots in the n-sphere we show that there is a relationship between the wrapped Floer cohomology algebra of the fiber and the Alexander invariant of the knot. This allows us to exhibit codimension two knots with infinite cyclic knot group such that the union of the unit conormal of the knot and the boundary of a cotangent fiber is not Legendrian isotopic to the union of the unit conormal of the unknot union the boundary and the same cotangent fiber. In Paper II we study the Chekanov-Eliashberg dg-algebra which is a holomorphic curve invariant associated to a smooth Legendrian submanifold. We extend this definition to singular Legendrians. Using the new definition we formulate and prove a surgery formula relating the wrapped Floer cohomology algebra of the co-core disk of a stop with the Chekanov- Eliashberg dg-algebra of its attaching locus interpreted as the Weinstein neighborhood of a singular Legendrian. A special case of this surgery formula, when the Legendrian is non- singular, establishes a proof of a conjecture by Ekholm-Lekili and independently by Sylvan. We furthermore provide an alternative geometric proof of the pushout diagrams for partially wrapped Floer cohomology and the stop removal formulas of Ganatra-Pardon-Shende. Keywords: Symplectic geometry, Contact geometry, Chekanov-Eliashberg dg-algebra, partially wrapped Floer cohomology, Lagrangian submanifold, Legendrian submanifold, singular Legendrians, based loop space Johan Asplund, Department of Mathematics, Box 480, Uppsala University, SE-75106 Uppsala, Sweden. © Johan Asplund 2021 ISSN 1401-2049 ISBN 978-91-506-2872-2 urn:nbn:se:uu:diva-439476 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-439476) List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I J. Asplund, Fiber Floer cohomology and conormal stops, arXiv:1912.02547 (2019). To appear in Journal of Symplectic Geometry. II J. Asplund and T. Ekholm, Chekanov–Eliashberg dg-algebras for singular Legendrians, arXiv:2102.04858 (2021). Submitted. Reprints were made with permission from the publishers. Contents 1 Introduction . ................................................................................................. 7 1.1 Brief overview of the results of the thesis ...................................... 7 1.2 History and motivation .................................................................... 8 1.3 Outline of the thesis ......................................................................... 8 2 Geometry .................................................................................................... 10 2.1 Symplectic geometry ..................................................................... 10 2.1.1 Background and some linear algebra ............................. 10 2.1.2 Basics on symplectic manifolds ..................................... 12 2.1.3 Submanifolds of symplectic manifolds .......................... 13 2.1.4 Hamiltonian dynamics .................................................... 14 2.2 Contact geometry ........................................................................... 15 2.2.1 Basics on contact manifolds ........................................... 16 2.2.2 Reeb dynamics ................................................................. 17 2.2.3 Submanifolds of contact manifolds ................................ 18 2.3 Liouville and Weinstein manifolds ............................................... 18 2.3.1 A connection between symplectic and contact geometry ........................................................................... 18 2.3.2 Weinstein manifolds ........................................................ 19 3 Gromov’s J-holomorphic curves, Floer theory and symplectic field theory .......................................................................................................... 22 3.1 J-holomorphic curves .................................................................... 22 3.2 Floer theory and the Arnold–Givental conjecture ........................ 24 3.2.1 Wrapped Floer cohomology ........................................... 26 3.3 Symplectic field theory .................................................................. 27 3.3.1 The Chekanov–Eliashberg dg-algebra ........................... 29 4 Summary of papers .................................................................................... 32 4.1 Paper I ............................................................................................. 32 4.2 Paper II ............................................................................................ 33 5 Summary in Swedish ................................................................................. 36 5.1 Artikel I ........................................................................................... 36 5.2 Artikel II ......................................................................................... 37 6 Acknowledgments ..................................................................................... 39 References ........................................................................................................ 40 1. Introduction In this chapter we first give a technical and brief overview of the results of the thesis, followed by history and motivation. For an introduction to the various mathematical concepts appearing in the thesis, we refer to Chapters 2 and 3. Chapter 5 consists of a summary in Swedish intended for non-technical readers. 1.1 Brief overview of the results of the thesis In this thesis we study partially wrapped Floer cohomology from the perspec- tives of contact geometry and symplectic field theory. Resulting from our work is a geometric description the endomorphism algebras of the generators of the partially wrapped Fukaya category of a stopped Weinstein manifold, in terms of the contact geometry of the attaching locus of the stop. In case the stopped Weinstein manifold is a cotangent bundle stopped by the unit conormal bundle of a submanifold, our description is in terms of contact geometry of the unit conormal bundle and chains of loops of the complement of the submanifold in the base together with the Pontryagin product. In Paper I we show that the wrapped Floer cohomology algebra of a cotan- gent fiber F in the stopped cotangent bundle is equivalent to chains of based loops on the complement of the submanifold in the base. For codimension 2 knots K in the n-sphere we show that there is a relationship between the wrapped Floer cohomology algebra of the fiber and the Alexander invariant of n ∼ K. This allows us to exhibit codimension 2 knots K satisfying π1(S nK) = Z and such that the Legendrian submanifold ΛK [@F is not Legendrian isotopic to Λunknot [ @F . In Paper II we study the Chekanov–Eliashberg dg-algebra which is a holo- morphic curve invariant associated to a smooth Legendrian submanifold. We extend this definition to singular Legendrians. Using the new definition we formulate and prove a surgery formula relating the wrapped Floer cohomology algebra of the co-core disk of a stop with the Chekanov–Eliashberg dg-algebra of its attaching locus interpreted as the Weinstein neighborhood of a singular Legendrian. A special case of this surgery formula, when the Legendrian is non-singular, establishes a proof of a conjecture by Ekholm–Lekili and inde- pendently by Sylvan [EL17, Syl19]. We furthermore provide an alternative geometric proof of the pushout diagrams for partially wrapped Floer cohomol- ogy and the stop removal formulas of Ganatra–Pardon–Shende [GPS18]. 7 1.2 History and motivation In physics, classical mechanics is the study of motion of macroscopic objects, such as projectiles or planets. Newton’s laws of motion describe the relation- ship between the motion of an object and the forces acting on it, and they form the foundation of Newtonian mechanics. A more sophisticated and general approach to classical mechanics is known as Hamiltonian mechanics and it is governed by Hamilton’s equations. The framework of Hamiltonian mechanics can also be used to describe quantum mechanics. Symplectic geometry forms the mathematical basis used to describe Hamilton’s equations and Hamilto- nian mechanics. Related to symplectic geometry is contact geometry. Contact geometry provides a mathematical framework which can be used to describe optical phenomena
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