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A connected sum formula for embedded contact homology

Luya Wang

TL;DR

This work proves a connected sum formula for embedded contact homology (ECH) by analyzing the chain-level behavior of ECH under Weinstein 1-handle connected sums. The authors isolate a special hyperbolic orbit $h$ and two planes $P_N$, $P_S$ in the symplectization to model the sum, then identify the filtered ECH of the connected sum with a filtered mapping cone Cone$^L$(\$U_1\otimes id + id\otimes U_2$) and construct a chain homotopy equivalence to the corresponding cone. Passing to direct limits yields the main result that $ECH(Y_1\#Y_2,\xi_1\#\xi_2,\Gamma_1+\Gamma_2) \cong ECH(Y_1,\xi_1,\Gamma_1) \tilde{\otimes}_{\\mathbb{F}[U]} ECH(Y_2,\xi_2,\Gamma_2)$, with a derived tensor product defined via the cone of the $U$-maps. This advances the understanding of ECH under connected sum, enabling computation of ECH for subcritical surgeries and suggesting extensions to related theories and to spectral-invariant refinements.

Abstract

Given two closed contact three-manifolds, one can form their contact connected sum via the Weinstein one-handle attachment. We study how pseudo-holomorphic curves in the symplectization behave under this operation. As a result, we give a connected sum formula for embedded contact homology.

A connected sum formula for embedded contact homology

TL;DR

This work proves a connected sum formula for embedded contact homology (ECH) by analyzing the chain-level behavior of ECH under Weinstein 1-handle connected sums. The authors isolate a special hyperbolic orbit and two planes , in the symplectization to model the sum, then identify the filtered ECH of the connected sum with a filtered mapping cone Cone(\) and construct a chain homotopy equivalence to the corresponding cone. Passing to direct limits yields the main result that , with a derived tensor product defined via the cone of the -maps. This advances the understanding of ECH under connected sum, enabling computation of ECH for subcritical surgeries and suggesting extensions to related theories and to spectral-invariant refinements.

Abstract

Given two closed contact three-manifolds, one can form their contact connected sum via the Weinstein one-handle attachment. We study how pseudo-holomorphic curves in the symplectization behave under this operation. As a result, we give a connected sum formula for embedded contact homology.
Paper Structure (28 sections, 34 theorems, 142 equations, 3 figures)

This paper contains 28 sections, 34 theorems, 142 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Embedded contact homology
  3. The main theorem
  4. Ideas of the proof of the main theorem
  5. Further directions
  6. An overview of ECH
  7. The Conley-Zehnder index
  8. The relative first Chern class
  9. The relative intersection pairing
  10. The Fredholm index and the ECH index
  11. The ECH differential and the U map
  12. The filtered ECH
  13. The connected sum for contact three-manifolds
  14. Weinstein 1-handle model and the Reeb flow
  15. Almost complex structures
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $(Y_1, \lambda_1)$ and $(Y_2, \lambda_2)$ be two closed connected contact three-manifolds with given nondegenerate contact forms $\lambda_1$ and $\lambda_2$. Then, as $\mathbb{F}$-modules.

Figures (3)

  • Figure 1: The base point $p$ on the pseudo-holomorphic plane $P_N$
  • Figure 2: An example of an $I = 2$ building that passes through $p$.
  • Figure 3: Schematic illustration of the proof involving direct limits in Theorem \ref{['thm:main']}. In the region above $R(L)$, i.e. when $r\leq R(L)$, we have isomorphisms of $ECH_*^{L}(Y_1\#_{r} Y_2, \alpha_{r})$ given a fixed value of $L$ by Proposition \ref{['prop:cobordism']}.

Theorems & Definitions (81)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Definition 1.5
  • Conjecture 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 71 more