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Noncommutativity of monotone Lagrangian cobordisms

Vardan Oganesyan

Abstract

We construct a monotone spin Lagrangian cobordism from L to (L_1, L_2) such that there is no monotone spin Lagrangian cobordism from L to (L_2, L_1), where L, L_1, L_2 are Lagrangians of CP^7.

Noncommutativity of monotone Lagrangian cobordisms

Abstract

We construct a monotone spin Lagrangian cobordism from L to (L_1, L_2) such that there is no monotone spin Lagrangian cobordism from L to (L_2, L_1), where L, L_1, L_2 are Lagrangians of CP^7.

Paper Structure

This paper contains 9 sections, 9 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lagrangian submanifolds of $\mathbb{C}P^n$
  4. Lagrangian cobordism
  5. Clean intersection and surgery
  6. Construction of noncommutative Lagrangian cobordisms
  7. Construction of the Lagrangian $L_1$
  8. Construction and properties of the cobordism
  9. Nonexistence of the cobordism $L \rightsquigarrow (L_2, L_1)$

Key Result

Theorem 2.1

(see Felix) Assume that - $L_1, L_2 \subset \mathbb{C}P^n$ are monotone Lagrangians and the minimal Maslov number of the pair $N_{L_1, L_2} > 3$ - $L_1, L_2$ are spin and we fix a relative spin structure - Let $\Lambda = \mathbb{Z}[T, T^{-1}]$ be the algebra of Laurent polynomials, where we grade $d It turns out that Let $N$ be a positive divisor of $N_{L_1, L_2}$ and $\widetilde{\Lambda} = \math

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 4 more