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Wall-crossing from Lagrangian Cobordisms

Jeff Hicks

Abstract

Biran and Cornea showed that monotone Lagrangian cobordisms give an equivalence of objects in the Fukaya category. However, there are currently no known non-trivial examples of monotone Lagrangian cobordisms with two ends. We look at an extension of their theory to the pearly model of Lagrangian Floer cohomology and unobstructed Lagrangian cobordisms. In particular, we examine the suspension cobordism of a Hamiltonian isotopy and the Haug mutation cobordism between mutant Lagrangian surfaces. In both cases, we show that these Lagrangian cobordisms can be unobstructed by bounding cochain and additionally induce an $A_\infty$ homomorphism between the Floer cohomology of the ends. This gives a first example of a two-ended Lagrangian cobordism giving a non-trivial equivalence of Lagrangian Floer cohomology. A brief computation is also included which shows that the incorporation of bounding cochain from this equivalence accounts for the "instanton-corrections" considered by Auroux, Ekholm, Pascaleff, Rizell, and Tonkonog for the wall-crossing formula between Chekanov and product tori in $(\mathbb C^2)\setminus \{z_1z_2=1\}$. We additionally prove some auxiliary results that may be of independent interest. These include a weakly filtered version of the Whitehead theorem for $A_\infty$ algebras and an extension of Charest-Woodward's stabilizing divisor model of Lagrangian Floer cohomology to Lagrangian cobordisms.

Wall-crossing from Lagrangian Cobordisms

Abstract

Biran and Cornea showed that monotone Lagrangian cobordisms give an equivalence of objects in the Fukaya category. However, there are currently no known non-trivial examples of monotone Lagrangian cobordisms with two ends. We look at an extension of their theory to the pearly model of Lagrangian Floer cohomology and unobstructed Lagrangian cobordisms. In particular, we examine the suspension cobordism of a Hamiltonian isotopy and the Haug mutation cobordism between mutant Lagrangian surfaces. In both cases, we show that these Lagrangian cobordisms can be unobstructed by bounding cochain and additionally induce an homomorphism between the Floer cohomology of the ends. This gives a first example of a two-ended Lagrangian cobordism giving a non-trivial equivalence of Lagrangian Floer cohomology. A brief computation is also included which shows that the incorporation of bounding cochain from this equivalence accounts for the "instanton-corrections" considered by Auroux, Ekholm, Pascaleff, Rizell, and Tonkonog for the wall-crossing formula between Chekanov and product tori in . We additionally prove some auxiliary results that may be of independent interest. These include a weakly filtered version of the Whitehead theorem for algebras and an extension of Charest-Woodward's stabilizing divisor model of Lagrangian Floer cohomology to Lagrangian cobordisms.

Paper Structure

This paper contains 49 sections, 32 theorems, 126 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Structure of the paper
  4. Notation
  5. Acknowledgements
  6. Background
  7. Lagrangian cobordisms
  8. Pearly model for Floer complex
  9. Domain and Label dependent perturbations
  10. Pearly Model for Lagrangian Cobordisms
  11. Toy model: continuation maps from cylindrical cobordisms
  12. Cobordisms and mutations
  13. Review of the Haug cobordism
  14. General mutations
  15. Geometric input
  16. ...and 34 more sections

Key Result

Theorem 1.1.1

The projections of the Floer cohomology of a Lagrangian cobordism to that of its ends $\beta^\pm: {CF^\bullet}(K)\to {CF^\bullet}(L^\pm)$ are $A_\infty$ homomorphisms.

Figures (13)

  • Figure 1: The profile of the Morse function for a cobordism. Bottlenecks are inserted on the ends of the cobordism.
  • Figure 2: The projection of the eye-shaped Lagrangian cobordism, $(K_r\circ K_l\cap U)\subset X\times \mathbb{C}\to \mathbb{C}$. The cycle $\pi^{-1}_{ U}(0,0)$ is highlighted in red.
  • Figure 3: All of the paths in the Lefschetz fibration needed to assemble the tori and cobordism
  • Figure 4: Steps to build our Lagrangian cobordism by projection to the cobordism $\mathbb{C}$ parameter of $X\times \mathbb{C}$.
  • Figure 5: We draw the sublevel sets of the Morse-Bott function for $K_{\mu_D}$ at the critical values of $\tilde{h}$. Notice that $L^0$ separates the cobordism into two halves.
  • ...and 8 more figures

Theorems & Definitions (98)

  • Theorem 1.1.1: Restatement of \ref{['cor:projectionsareainfinity']}
  • Theorem 1.1.2: Paraphrasing \ref{['thm:cylindricityofcobordism']}
  • Theorem 1.1.3: Paraphrasing \ref{['thm:wallcrossingcobordism']}
  • Definition 2.1.1: arnol1980lagrange
  • Theorem 2.1.2
  • Definition 2.1.3
  • Proposition 2.1.4
  • Theorem 2.2.1: charest2015floer
  • Lemma 2.2.2: Paraphrasing \ref{['lemma:geometricflowlines']}
  • Lemma 2.2.3: Paraphrasing of \ref{['lem:weaklystabilizedcobordism', 'lem:estabilizedcobordism']}
  • ...and 88 more