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Mapping tori of $A_{\infty}$-autoequivalences and Legendrian lifts of exact Lagrangians in circular contactizations

Adrian Petr

Abstract

We study mapping tori of quasi-autoequivalences $τ: \mathcal{A} \to \mathcal{A}$ which induce a free action of $\mathbf{Z}$ on objects. More precisely, we compute the mapping torus of $τ$ when it is strict and acts bijectively on hom-sets, or when the $A_{\infty}$-category $\mathcal{A}$ is directed and there is a bimodule map $\mathcal{A} (-, -) \to \mathcal{A} (-, τ(-))$ satisfying some hypotheses. Then we apply these results in order to link together the Fukaya $A_{\infty}$-category of a family of exact Lagrangians, and the Chekanov-Eliashberg DG-category of Legendrian lifts in the circular contactization.

Mapping tori of $A_{\infty}$-autoequivalences and Legendrian lifts of exact Lagrangians in circular contactizations

Abstract

We study mapping tori of quasi-autoequivalences which induce a free action of on objects. More precisely, we compute the mapping torus of when it is strict and acts bijectively on hom-sets, or when the -category is directed and there is a bimodule map satisfying some hypotheses. Then we apply these results in order to link together the Fukaya -category of a family of exact Lagrangians, and the Chekanov-Eliashberg DG-category of Legendrian lifts in the circular contactization.
Paper Structure (62 sections, 55 theorems, 280 equations, 4 figures)

This paper contains 62 sections, 55 theorems, 280 equations, 4 figures.

Table of Contents

  1. Algebra.
  2. Mapping torus of an $A_{\infty}$-autoequivalence.
  3. Chekanov-Eliashberg DG-algebra.
  4. Legendrian lifts of exact Lagrangians in the circular contactization.
  5. Acknowledgments.
  6. Algebra
  7. Modules over $A_{\infty}$-categories
  8. Pullback of $A_{\infty}$-modules.
  9. Quotient of $A_{\infty}$-modules.
  10. Relations between pullback and quotient of $A_{\infty}$-modules.
  11. Cone of module maps.
  12. Grothendieck construction and homotopy pushout
  13. Cylinder object and homotopy
  14. Adjunctions between Adams-graded and non Adams-graded
  15. Mapping torus of an $A_{\infty}$-autoequivalence
  16. ...and 47 more sections

Key Result

Theorem A

Let $\tau$ be a quasi-autoequivalence of an $A_{\infty}$-category $\mathcal{A}$, weakly directed with respect to some compatible $\mathbf{Z}$-splitting of $\mathrm{ob} \left( \mathcal{A} \right)$. Assume that there exists a closed degree $0$ bimodule map $f : \mathcal{A}_m \left( -, - \right) \to \m

Figures (4)

  • Figure 1: Reeb chords (in blue) of $\Lambda^{\circ} = \{ 0 \} \times 0_{S^1}$ for $\alpha^{\circ}$ (on the left) and for $\alpha_H^{\circ}$ (on the right)
  • Figure 2: Action of the projection $\Pi_{S^1 \times T^* S^1}$
  • Figure 3: Action of the contactomorphism $\phi_H^{-1}$
  • Figure 4: Action of the projection $\Pi_{T^*S^1}$

Theorems & Definitions (167)

  • Theorem A
  • Theorem B
  • Remark
  • Corollary
  • Definition 1.1
  • Definition 1.2
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • ...and 157 more