Embedded Contact Homology, the Weinstein Conjecture, and the Seiberg–Witten¶
Job: develop the ECH (Hutchings, embedded Reeb orbits / holomorphic curves), the Hutchings–Taubes isomorphism ECH \(\cong\) SW Floer, the Weinstein conjecture via ECH (Taubes 3D), the ECH capacities / symplectic embedding, and the Seiberg–Witten / Heegaard Floer contact invariant — with the ECH \(\cong\) SW Floer / the Weinstein in 3D as the worked example.
Core¶
Embedded contact homology (ECH) and the Hutchings–Taubes isomorphism¶
The ECH (the Hutchings: the homology of the contact 3-manifold \((Y,\xi)\) generated by the embedded Reeb orbits, the ECH chain complex, the differential counting the holomorphic curves / the embedded).
Theorem (Hutchings–Taubes). \(\mathrm{ECH}_*(Y,\xi)\cong\widehat{\mathrm{HM}}_*(-Y)\) (the ECH the Seiberg–Witten Floer homology, the Taubes).
| ECH / Hutchings–Taubes | Description |
|---|---|
| ECH | the Hutchings: the embedded Reeb orbits / the holomorphic curves (the ECH chain complex) |
| Embedded Reeb orbits | the generators (the embedded, the orbit sets, the ECH the embedded) |
| Holomorphic curves | the differential (the holomorphic curves in the symplectization, the embedded) |
| Hutchings–Taubes | \(\mathrm{ECH}\cong\widehat{\mathrm{HM}}\) (the ECH the Seiberg–Witten Floer, the Taubes) |
The Weinstein conjecture via ECH¶
| Weinstein / ECH / volume | Description |
|---|---|
| Weinstein conjecture | every contact form on a closed 3-manifold has a closed Reeb orbit (the Taubes 2007) |
| ECH proof | the ECH the nontrivial (\(\mathrm{ECH}\ne0\), the \(\widehat{\mathrm{HM}}\ne0\)) \(\Rightarrow\) the closed Reeb orbits (the generator) |
| Contact volume | \(\int\alpha\wedge d\alpha\) (the contact volume, the ECH the volume / the spectrum) |
The ECH and the symplectic embedding / the capacity¶
| ECH capacities / embedding | Description |
|---|---|
| ECH capacities (Hutchings) | the ECH the contact / symplectic capacity (obstructs the symplectic embeddings) |
| Symplectic embedding | the ECH the obstruction to the symplectic ball embeddings (the sharp for the 4D) |
| 4D sharp / knot filter | the ECH the sharp for the 4D embeddings (the knot filtration / the ECH the capacity) |
The Seiberg–Witten Floer and the contact invariant¶
| SW Floer / contact invariant / HF | Description |
|---|---|
| Seiberg–Witten Floer \(\widehat{\mathrm{HM}}\) | the monopole Floer (the Kronheimer–Mrowka) |
| Contact invariant \(c(\xi)\) | \(c(\xi)\in\widehat{\mathrm{HM}}(-Y)\) (the Ozsváth–Szabó, the contact the tight / the overtwisted the vanishing) |
| Heegaard Floer / contact | the Ozsváth–Szabó, the contact invariant (the tight the nonzero, the overtwisted the zero) |
The ECH / the SFT and the relation to the Heegaard Floer¶
| ECH-HM / HF / contact invariants | Description |
|---|---|
| ECH \(\cong\) HM (Taubes) | the ECH the Seiberg–Witten Floer (the equivalence) |
| Heegaard Floer / ECH | the \(\widehat{\mathrm{HF}}\cong\mathrm{ECH}\) (the Kutluhan–Lee–Taubes / the Colin–Ghiggini–Honda) |
| Contact invariants | the ECH / the HF / the SW the contact invariant (the equivalence) |
See it¶
Hutchings–Taubes ECH \(\cong\) Seiberg–Witten Floer and the Weinstein conjecture in dimension 3. The ECH \(\mathrm{ECH}_*(Y,\xi)\) (the embedded Reeb orbits / the holomorphic curves, the Hutchings). The Hutchings–Taubes: \(\mathrm{ECH}_*(Y,\xi)\cong\widehat{\mathrm{HM}}_*(-Y)\) (the ECH the Seiberg–Witten Floer, the Taubes the isomorphism, the nontrivial the \(\widehat{\mathrm{HM}}\ne0\)). The Weinstein: the ECH the nontrivial (the \(\mathrm{ECH}\ne0\), the \(\widehat{\mathrm{HM}}\ne0\)) \(\Rightarrow\) the closed Reeb orbits (the generator of the ECH the closed orbits, the ECH the Reeb orbits the generator), the Weinstein the consequence (the closed Reeb orbits exist, the Taubes 2007). The key point: the ECH the Seiberg–Witten Floer (the Hutchings–Taubes), the nontriviality the closed Reeb orbits, the Weinstein conjecture in 3D the Taubes — the ECH / the Seiberg–Witten the proof of the Weinstein, the contact / the gauge-theoretic.
Check¶
Show that the ECH the nontrivial (\(\mathrm{ECH}\ne0\)) implies the closed Reeb orbits, and the Weinstein.
The ECH \(\mathrm{ECH}_*(Y,\xi)\) (the embedded Reeb orbits / the holomorphic curves). The Hutchings–Taubes isomorphism: \(\mathrm{ECH}_*(Y,\xi)\cong\widehat{\mathrm{HM}}_*(-Y)\) (the ECH the Seiberg–Witten Floer). The \(\widehat{\mathrm{HM}}_*(-Y)\ne0\) (the Seiberg–Witten Floer the nontrivial for the closed 3-manifold \(-Y\), the Kronheimer–Mrowka — the \(\widehat{\mathrm{HM}}\) the nonvanishing for the contact / the nontrivial). The \(\mathrm{ECH}\ne0\) (the \(\cong\) the \(\widehat{\mathrm{HM}}\ne0\)) \(\Rightarrow\) the ECH the nontrivial \(\Rightarrow\) the closed Reeb orbits: the ECH the chain complex generated by the embedded Reeb orbit sets, the nontrivial ECH \(\Rightarrow\) the chain complex the nontrivial \(\Rightarrow\) the generators exist (the embedded Reeb orbit sets) \(\Rightarrow\) the closed Reeb orbits exist (the generator of the ECH the closed orbits). The Weinstein: the closed Reeb orbits exist (the ECH the nontrivial \(\Rightarrow\) the orbits), the Weinstein the consequence (the Taubes 2007).
Explain the ECH capacities and the symplectic embedding obstruction, and the 4D sharp.
The ECH capacities (Hutchings): the ECH defines a sequence of capacities \(c_k(Y,\xi)\) (the \(k\)-th ECH capacity, from the ECH the filtration / the action spectrum, the \(c_k\) the \(k\)-th level of the ECH). The symplectic embedding: the ECH capacities obstruct the symplectic embeddings — if \((Y_1,\xi_1)\) symplectically embeds into \((Y_2,\xi_2)\), then \(c_k(Y_1,\xi_1)\le c_k(Y_2,\xi_2)\) for all \(k\) (the monotonicity, the ECH the obstruction). The 4D sharp: the ECH capacities are the sharp for the 4D ball embeddings (the \(c_k\) the sharp obstruction for the symplectic ball embeddings into the 4D domains, the Hutchings — the ECH the sharp for the 4D, the higher-D the open). The knot filter: the ECH the knot filtration (the knot in the contact, the ECH the filtered, the capacity the knot-filtered). So the ECH the symplectic embedding obstruction (the sharp for the 4D), the capacity the ECH the obstruction.
Pitfalls¶
- The ECH the Hutchings–Taubes (the ECH \(\cong\) SW Floer, the nontriviality the Weinstein) — the ECH the Seiberg–Witten Floer (the Taubes), the nontriviality the closed Reeb orbits, the Weinstein the consequence.
- The ECH capacities the sharp for the 4D (the higher-D the open) — the ECH the sharp for the 4D ball embeddings, the higher-D the open / partial.
- The contact invariant \(c(\xi)\) the tight the nonzero, the overtwisted the zero (the HF / the SW the contact) — the tight the \(c(\xi)\ne0\), the overtwoted the \(c(\xi)=0\) (the Ozsváth–Szabó, the Kronheimer–Mrowka).
- The ECH \(\cong\) HM (Taubes) the equivalence (the ECH the SW Floer), the Heegaard Floer \(\cong\) ECH (the Kutluhan–Lee–Taubes / the Colin–Ghiggini–Honda) — the three the equivalent (the ECH / the HM / the HF), the contact invariants the same.
- The Weinstein the Taubes (the 3D, the Seiberg–Witten / the ECH, the higher-D the open) — the 3D the proven (the Taubes), the higher-D the open.
Refs¶
- Hutchings, "Embedded contact homology and its applications" (2010); Hutchings–Taubes, "Proof of the Weinstein conjecture via ECH" (the isomorphism)
- Taubes, "Embedded contact homology and Seiberg–Witten Floer homology" (J. Topol. 2010, the \(\mathrm{ECH}\cong\widehat{\mathrm{HM}}\))
- Kronheimer–Mrowka, Monopoles and Three-Manifolds (the Seiberg–Witten Floer); Ozsváth–Szabó, the Heegaard Floer / the contact invariant
- Kutluhan–Lee–Taubes, "HF = ECH" (2010); Colin–Ghiggini–Honda, "Equivariant HF and ECH" (2011)
Next¶
CG index · Foundations · Legendrian & Transverse · Contact Homology & SFT · h-Principle & Flexibility · Symplectic / Floer · Topology