The h-Principle, Overtwisted/Loose, and Flexible vs Rigid¶
Job: develop the Gromov / Eliashberg \(h\)-principle (open/ample), the overtwisted (3D) / loose (higher-D) classification (Eliashberg / BEM), the Weinstein conjecture / Reeb orbits (Taubes 3D / higher-D open), and the flexible Weinstein cobordism — with the Eliashberg \(h\)-principle for overtwisted / the Taubes Weinstein as the worked example.
Core¶
The \(h\)-principle (Gromov / Eliashberg)¶
The \(h\)-principle (the Gromov: the differential relations the formal / the homotopy-theoretic solutions \(\Leftrightarrow\) the genuine, the flexible).
| \(h\)-principle / open-ample | Description |
|---|---|
| \(h\)-principle | the formal / homotopy-theoretic solutions \(\Leftrightarrow\) the genuine (the flexible) |
| Open / ample | the contact / the symplectic / the isosymplectic (the open conditions the \(h\)-principle) |
| Eliashberg \(h\)-principle (overtwisted) | the 3D: the overtwisted contact structures classified by the homotopy of the plane fields (the formal = the genuine) |
| Formal / genuine | the formal contact structure (the plane field) vs the genuine (the contact), the \(h\)-principle the equivalence |
The overtwisted (3D) and the loose (higher-D)¶
| Overtwisted / loose / tight | Description |
|---|---|
| Overtwisted (3D) | the overtwisted disk, the Eliashberg classification by the homotopy of \(\xi\) as a plane field, the \(h\)-principle (the flexible) |
| Loose (higher-D) | the Borman–Eliashberg–Murphy the loose chart, the loose contact the \(h\)-principle (the flexible) |
| Tight / non-loose | the rigid, the non-\(h\)-principle, the Bennequin / the Chekanov (the rigidity) |
| 3D / higher-D | the 3D the Eliashberg (overtwisted), the higher-D the BEM (loose) |
The Borman–Eliashberg–Murphy (BEM) and the higher-D \(h\)-principle¶
Theorem (BEM). In dimensions \(\ge5\), the loose contact structures are classified by the formal contact structures (the homotopy-theoretic, the \(h\)-principle).
| BEM / loose chart / higher-D | Description |
|---|---|
| BEM theorem | the loose contact (\(\ge5\)) classified by the formal (the \(h\)-principle) |
| Loose chart | the local model (the flexible, the overtwisted the 3D analog) |
| Flexible vs rigid | the flexible the \(h\)-principle, the rigid the holomorphic curves (the dichotomy) |
The Weinstein conjecture and the existence of Reeb orbits¶
| Weinstein / Reeb orbits | Description |
|---|---|
| Weinstein conjecture | every contact form on a closed contact manifold has at least one closed Reeb orbit |
| 3D (Taubes) | the Weinstein in dim 3 proved by Taubes (via the Seiberg–Witten = the ECH) |
| Higher-D | the open / the partial (the fillable, the \(S^1\)-invariant, the Hofer) |
| ECH and the Weinstein | the Hutchings–Taubes: the ECH the proof of the Weinstein in 3D |
The flexible Weinstein and the cobordism¶
| Flexible Weinstein / cobordism | Description |
|---|---|
| Flexible Weinstein (Cieliebak–Eliashberg) | the flexible Weinstein cobordisms the \(h\)-principle (the subcritical / the flexible) |
| Weinstein / flexible vs rigid | the flexible the \(h\)-principle, the rigid the holomorphic curves (the Murphy) |
| Cobordism / surgery | the contact surgery, the Weinstein handle (the flexible / the subcritical) |
See it¶
Eliashberg \(h\)-principle for the overtwisted contact structures on 3-manifolds and the Taubes Weinstein conjecture. The overtwisted contact structures on a 3-manifold \(Y\) (the overtwisted disk \(D_{\mathrm{ot}}\), the Eliashberg 1989: the overtwisted contact structures on \(Y\) are classified by the homotopy classes of the oriented 2-plane fields \(\xi\subset TY\), the formal contact structures — the \(h\)-principle, the formal = the genuine). The tight (the non-overtwisted, the rigid: the Bennequin inequality, the Eliashberg–Gromov the non-\(h\)-principle, the tight the holomorphic-curve rigidity). The Weinstein conjecture (the Taubes 2007: every contact form on a closed 3-manifold has a closed Reeb orbit, the proof via the Seiberg–Witten = the ECH, the embedded contact homology, the Hutchings–Taubes the isomorphism ECH \(\cong\) Seiberg–Witten Floer). The key point: the overtwisted the \(h\)-principle (the flexible, the homotopy the classification), the tight the rigid (the holomorphic curves, the Bennequin), the Weinstein the Taubes (the Seiberg–Witten / the ECH) — the flexible / the rigid, the \(h\)-principle / the holomorphic-curve rigidity.
Check¶
Show that the overtwisted contact structures on a 3-manifold \(Y\) are classified by the homotopy of the plane fields (the Eliashberg \(h\)-principle), and explain the tight vs the overtwisted.
The overtwisted: a contact structure \(\xi\) on \(Y^3\) is overtwisted if it contains an overtwisted disk \(D_{\mathrm{ot}}\) (a disk where \(\xi\) is tangent to the boundary and twists once). The Eliashberg \(h\)-principle (1989): the overtwisted contact structures on \(Y\) are in bijection with the homotopy classes of the oriented 2-plane fields \(\xi\subset TY\) (the formal contact structures — the plane fields, no contact condition). The \(h\)-principle: the formal (the plane field) = the genuine (the overtwisted contact), the classification the homotopy-theoretic. The tight: the non-overtwisted, the rigid — the Bennequin inequality (the \(\mathrm{tb}+|\mathrm{rot}|\le-\chi\)), the Eliashberg–Gromov the non-\(h\)-principle (the tight the holomorphic-curve rigidity, the no-overtwisted-disk the rigidity). The tight vs the overtwisted: the tight the rigid (the holomorphic curves), the overtwisted the flexible (the \(h\)-principle, the homotopy).
Explain the Weinstein conjecture and the Taubes proof in 3D, and the status in higher dimensions.
The Weinstein conjecture: every contact form \(\alpha\) on a closed contact manifold \((Y,\xi)\) has at least one closed Reeb orbit (a periodic orbit of the Reeb flow \(R_\alpha\)). The 3D (Taubes 2007): the Weinstein in dim 3 proved by Taubes, via the Seiberg–Witten Floer / the ECH (the embedded contact homology, the Hutchings–Taubes isomorphism \(\mathrm{ECH}\cong\widehat{\mathrm{HM}}\)). The mechanism: the ECH the nontrivial (\(\mathrm{ECH}\ne0\) the \(\widehat{\mathrm{HM}}\ne0\)), the nontrivial ECH \(\Rightarrow\) the closed Reeb orbits (the generator of the ECH the closed orbits), the Weinstein the consequence. The higher-D: the open / the partial — the fillable (the symplectically fillable the Weinstein the Hofer), the \(S^1\)-invariant (the \(S^1\)-invariant contact form the Weinstein), the general the open.
Pitfalls¶
- The overtwisted the \(h\)-principle (the flexible, the formal = the genuine), the tight the rigid (the holomorphic curves, the Bennequin) — the overtwisted the \(h\)-principle, the tight the non-\(h\)-principle / the rigid.
- The BEM the higher-D (the 3D the Eliashberg, the higher-D the BEM) — the 3D the Eliashberg overtwisted, the higher-D the BEM loose.
- The Weinstein the Taubes (the 3D, the Seiberg–Witten / the ECH, the higher-D the open) — the 3D the proven, the higher-D the open / partial.
- The flexible Weinstein the \(h\)-principle (the subcritical / the flexible), the rigid the holomorphic (the critical / the rigid) — the subcritical the flexible, the critical the rigid.
- The loose chart the local model (the flexible, the overtwisted the 3D analog) — the loose the higher-D \(h\)-principle, the overtwisted the 3D \(h\)-principle.
Refs¶
- Eliashberg, "Classification of overtwisted contact structures on 3-manifolds" (Invent. 1989)
- Gromov, Partial Differential Relations (the \(h\)-principle); Borman–Eliashberg–Murphy, "Existence and classification of overtwisted contact structures in higher dimensions" (2015)
- Taubes, "The Seiberg–Witten equations and the Weinstein conjecture" (2007); the ECH / the Weinstein
- Cieliebak–Eliashberg, From Stein to Weinstein and Back (the flexible Weinstein, the cobordism)
Next¶
CG index · Foundations · Legendrian & Transverse · Contact Homology & SFT · ECH & Weinstein · Symplectic / Floer · Differential geometry