Reflections on cylindrical contact homology Jo Nelson (Rice) Symplectic Zoominar, May 2020 https://math.rice.edu/~jkn3/Zoominar-slides.pdf Jo Nelson (Rice) Reflections on cylindrical contact homology
Contact structures Definition A contact structure is a maximally nonintegrable hyperplane field. ξ = ker( dz − ydx ) The kernel of a 1-form λ on Y 2 n − 1 is a contact structure whenever λ ∧ ( d λ ) n − 1 is a volume form ⇔ d λ | ξ is nondegenerate. Jo Nelson (Rice) Reflections on cylindrical contact homology
Reeb vector fields Definition The Reeb vector field R on ( Y , λ ) is uniquely determined by λ ( R ) = 1 , d λ ( R , · ) = 0. The Reeb flow , ϕ t : Y → Y is defined by d dt ϕ t ( x ) = R ( ϕ t ( x )). A closed Reeb orbit (modulo reparametrization) satisfies γ : R / T Z → Y , γ ( t ) = R ( γ ( t )) , ˙ (0.1) and is embedded whenever (0.1) is injective. Jo Nelson (Rice) Reflections on cylindrical contact homology
Reeb orbits Given an embedded Reeb orbit γ : R / T Z → Y , the linearized flow along γ defines a symplectic linear map d ϕ t : ( ξ | γ (0) , d λ ) → ( ξ | γ ( t ) , d λ ) d ϕ T is called the linearized return map . If 1 is not an eigenvalue of d ϕ T then γ is nondegenerate . λ is nondegenerate if all Reeb orbits associated to λ are nondegenerate. In dim 3, nondegenerate orbits are either elliptic or hyperbolic according to whether d ϕ T has eigenvalues on S 1 or real eigenvalues. Jo Nelson (Rice) Reflections on cylindrical contact homology
Reeb orbits on S 3 S 3 := { ( u , v ) ∈ C 2 | | u | 2 + | v | 2 = 1 } , λ = i 2 ( ud ¯ u − ¯ udu + vd ¯ v − ¯ vdv ) . The orbits of the Reeb vector field form the Hopf fibration! R = iu ∂ u ∂ u + iv ∂ v ∂ ∂ u − i ¯ ∂ v − i ¯ v = ( iu , iv ) . ∂ ¯ ∂ ¯ The flow is ϕ t ( u , v ) = ( e it u , e it v ). Niles Johnson, S 3 / S 1 = S 2 Patrick Massot Jo Nelson (Rice) Reflections on cylindrical contact homology
A video of the Hopf fibration Jo Nelson (Rice) Reflections on cylindrical contact homology
A new era of contact geometry Helmut Hofer on the origins of the field: So why did I come into symplectic and contact geom- etry? So it turned out I had the flu and the only thing to read was a copy of Rabinowitz’s paper where he proves the existence of periodic orbits on star-shaped energy sur- faces. It turned out to contain a fundamental new idea, which was to study a different action functional for loops in the phase space rather than for Lagrangians in the con- figuration space. Which actually if we look back, led to the variational approach in symplectic and contact topology, which is reincarnated in infinite dimensions in Floer the- ory and has appeared in every other subsequent approach. ...Ja, the flu turned out to be really good. Jo Nelson (Rice) Reflections on cylindrical contact homology
Existence of periodic orbits The Weinstein Conjecture (1978) Let Y be a closed oriented odd-dimensional manifold with a contact form λ . Then the associated Reeb vector field R admits a closed orbit. Weinstein (convex hypersurfaces) Rabinowitz (star shaped hypersurfaces) Star shaped is secretly contact! Viterbo, Hofer, Floer, Zehnder (‘80’s fun) Hofer ( S 3 ) Taubes (dimension 3) Tools > 1985: Floer Theory and Gromov’s pseudoholomorphic curves. Jo Nelson (Rice) Reflections on cylindrical contact homology
The machinery that was invented Let ( Y 2 n − 1 , ξ = ker λ ) be a closed nondegenerate contact manifold. Floerify Morse theory on C ∞ ( S 1 , Y ) A : → R , ż �→ γ λ. γ Proposition γ ∈ Crit( A ) ⇔ γ is a closed Reeb orbit. Grading: | γ | = CZ ( γ ) + n − 3, C EGH ( Y , λ, J ) = Q �{ closed Reeb orbits } \ { bad Reeb orbits }� ∗ 3-D : Even covers of embedded negative hyperbolic orbits are bad. Jo Nelson (Rice) Reflections on cylindrical contact homology
The letter J is for pseudoholomorphic A λ -compatible almost complex structure is a J on T ( R × Y ): J is R -invariant J ξ = ξ , positively with respect to d λ J ( ∂ s ) = R , where s denotes the R coordinate Gradient flow lines are a no go; instead count pseudoholomorphic cylinders u ∈ M J ( γ + , γ − ) / R . s →±∞ π R u ( s , t ) = ±∞ lim u : ( R × S 1 , j ) → ( R × Y , J ) s →±∞ π M u ( s , t ) = γ ± lim ¯ ∂ J u := du + J ◦ du ◦ j ≡ 0 up to reparametrization . Note: J is S 1 -INDEPENDENT Jo Nelson (Rice) Reflections on cylindrical contact homology
Cylindrical contact homology C ∗ ( Y , λ, J ) = Q �{ closed } \ { bad }� | γ | = CZ ( γ ) + n − 3 α = γ kp γ p + + ÿ m ( α ) � ∂ EGH α, β � = k :1 m ( u ) ǫ ( u ) − → u ∈M J ( α,β ) / R , β = γ kq γ q | α |−| β | =1 − − ÿ m ( β ) γ ± embedded, gcd( p , q ) = 1 EGH � ∂ α, β � = m ( u ) ǫ ( u ) u ∈M J ( α,β ) / R , | α |−| β | =1 Conjecture (Eliashberg-Givental-Hofer ’00) If there are no contractible Reeb orbits with | γ | = − 1 , 0 , 1 then ( C ∗ , ∂ ) is a chain complex and CH EGH ( Y , ker λ ; Q ) = H ( C ∗ ( Y , λ, J ) , ∂ ) is an ∗ invariant of ξ = ker λ . Jo Nelson (Rice) Reflections on cylindrical contact homology
Progress... Definition ( Y 2 n +1 , λ ) is hypertight if there are no contractible Reeb orbits. ( Y 3 , λ ) is dynamically convex whenever c 1 ( ξ ) | π 2 ( Y ) = 0 and every contractible γ satisfies CZ ( γ ) ≥ 3 . For us { hypertight } ⊂ { dynamically convex } . A convex hypersurface transverse to the radial vector field Y in ( R 4 , ω 0 ) admits a dynamically convex contact form λ 0 := ω 0 ( Y , · ). Theorem (Hutchings-N. ‘14 (JSG 2016)) If ( Y 3 , λ ) is dynamically convex, J generic, and every contractible Reeb orbit γ has CZ ( γ ) = 3 only if γ is embedded then ∂ 2 = 0 . Intersection theory is key to our proof that ∂ 2 = 0. Can allow contractible CZ ( γ ) = 3 for prime covers of embedded Reeb orbits (Cristofaro-Gardiner - Hutchings - Zhang) 3D hypertight: invariance via obg for chain homotopy (Bao - Honda ’14) Any dim hypertight: ∂ 2 = 0 and invariance via Kuranishi atlases (Pardon ’15) Jo Nelson (Rice) Reflections on cylindrical contact homology
The pseudoholomorphic menace Transversality for multiply covered curves is hard. Is M J ( γ + ; γ − ) more than a set? M J ( γ + ; γ − ) can have nonpositive virtual dimension... Compactness issues are “severe”. − 3 1 ind= 2 2 − 1 0 1 0 2 2 Desired compactification Adding to 2 becomes hard when CZ ( x ) − CZ ( z ) = 2. Jo Nelson (Rice) Reflections on cylindrical contact homology
The return of regularity (domain dependent J ) S 1 -independent J cylinders in R × Y 3 are reasonable All hope is lost in cobordisms, and no obvious chain maps. ( Y , λ, J ) requires S 1 -dependent J := { J t } t ∈ S 1 . Invariance of CH EGH ∗ ∂ EGH � 2 = 0. But breaking S 1 -symmetry invalidates � We define a Morse-Bott non-equivariant chain complex à q ∂ + ∂ ∂ NCH := NCC ∗ := Z � q γ, p γ � , ∂ − + obg p ∂ all Reeb orbits γ If sufficient regularity exists to use J , then between good orbits, ∂ + = 0 q p EGH ∂ = ∂ EGH , ∂ = − ∂ , Compactness issues require obstruction bundle gluing, producing a novel correction term. The nonequivariant theory NCH ∗ is a contact invariant, which we relate to CH EGH via family Floer methods. ∗ Jo Nelson (Rice) Reflections on cylindrical contact homology
Enter the point constraints Given a generic λ -compatible family J := { J t } t ∈ S 1 , p α α M J ( γ + , γ − ) e ± : → im( γ ± ) = γ ± ← u ( R ×{ 0 } ) u �→ s →±∞ π Y u ( s , 0) lim Can use to specify a generic base point p γ on each β embedded γ : e + ( u ) = p α , e − ( u ) = p β . p β The base level cascade Morse-Bott moduli spaces , M J ( · , · ) 1 : M J � � α, q p β := M J ( α, β ) 1 M J � � α, q � � q β := u ∈ M J ( α, β ) | e + ( u ) = p α 1 M J � � � � α, p p β := u ∈ M J ( α, β ) | e − ( u ) = p β 1 M J � � � � α, p q β := u ∈ M J ( α, β ) | e + ( u ) = p α , e − ( u ) = p β 1 Higher levels consist of certain tuples ( u 1 , .., u ℓ ) of broken cylinders. As a set, each of these spaces is a disjoint union of subsets M J ( · , · ) ℓ . Jo Nelson (Rice) Reflections on cylindrical contact homology
Enter the cascade moduli spaces Definition Assuming α = γ 0 , γ 1 , ... , γ ℓ − 1 , γ ℓ = β are all distinct, the � � α, r higher levels M J r β ℓ , are the set of tuples | α |−| β | ź ℓ M J ( γ i − 1 , γ i ) ( u 1 , .., u ℓ ) ∈ such that i =1 if r α = q α then e + ( u 0 ) = p α ; if r β = p β then e − ( u ℓ ) = p β ; e − ( u i − 1 ), e + ( u i ), p γ i , are cyclically ordered wrt Reeb M J α, p 3 ( q β ) flow. When α = β , define M J ( q α ) = M J ( q α ) = M J ( p α ; q α ; p α ; p α ) = ∅ , − 2 { pt } if α is bad; M J ( p α ; q α ) := ∅ if α is good. Jo Nelson (Rice) Reflections on cylindrical contact homology
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