Embedded Contact Homology of Prequantization Bundles Jo Nelson - PowerPoint PPT Presentation

Embedded Contact Homology of Prequantization Bundles Jo Nelson & Morgan Weiler Rice University WHVSS, May 2020 https://math.rice.edu/~jkn3/WHVSS-slides.pdf Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Contact structures Definition A contact structure is a maximally nonintegrable hyperplane field. ξ = ker( dz − ydx ) Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

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 & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

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 ( ?? ) is injective. Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

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 . Nondegenerate orbits are either elliptic or hyperbolic according to whether d ϕ T has eigenvalues on S 1 or real eigenvalues. λ is nondegenerate if all Reeb orbits associated to λ are nondegenerate. Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

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 & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

A video of the Hopf fibration Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Prequantization bundles Theorem (Boothby-Wang construction ’58) Let (Σ g , ω ) be a Riemann surface and e a negative class in H 2 (Σ g ; Z ) . Let p : Y → Σ g be the principal S 1 -bundle with Euler class e. Then there is a connection 1-form λ on Y whose Reeb vector field R is tangent to the S 1 -action. ( Y , λ ) is the prequantization bundle over (Σ g , ω ). The Reeb orbits of R are the S 1 -fibers of this bundle. The Reeb orbits of R are degenerate. d λ = p ∗ ω p ∗ ξ = T Σ g Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Perturbed Reeb dynamics of prequantization bundles Use a Morse-Smale H : Σ → R , | H | C 2 < 1 to perturb λ : λ ε := (1 + ε p ∗ H ) λ The perturbed Reeb vector field is ε ˜ R X H R ε = 1 + ε p ∗ H + (1 + ε p ∗ H ) 2 where ˜ X H is the horizontal lift of X H to ξ . If p ∈ Crit( H ) then X H ( p ) = 0. The action of a closed orbit γ is A ( γ ) := � γ λ ε . Fix L > 0. ∃ ε > 0 such that if γ is an orbit of R ǫ and if A ( γ ) < L then γ is nondegenerate and projects to p ∈ Crit( H ); if A ( γ ) > L then γ loops around the tori above the orbits of X H , or is a larger iterate of a fiber above p ∈ Crit( H ). Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Fiber orbits of prequantization bundles Recall ε ˜ R X H R ε = 1 + ε p ∗ H + (1 + ε p ∗ H ) 2 Denote the k -fold cover projecting to p ∈ Crit( H ) by γ k p . We have p ) = RS τ (fiber k ) − dim(Σ) CZ τ ( γ k + ind p ( H ) . 2 Using the constant trivialization of ξ = p ∗ T Σ, RS τ (fiber k ) = 0. Thus CZ τ ( γ k p ) = ind p ( H ) − 1 . Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Fiber orbits of prequantization bundles Recall CZ τ ( γ k p ) = ind p ( H ) − 1 Only positive hyperbolic orbits have even CZ . If ind p ( H ) = 1 then γ p is positive hyperbolic. Since p is a bundle, all linearized return maps are close to Id . Hence no negative hyperbolic orbits. If ind p ( H ) = 0 , 2 then γ p is elliptic. Assume H is perfect. Denote the index zero elliptic orbit by e − the index two elliptic orbit by e + , the hyperbolic orbits by h 1 , . . . , h 2 g . Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Embedded contact homology Embedded contact homology (ECH) is a Floer theory for closed ( Y 3 , λ ) and Γ ∈ H 1 ( Y ; Z ). For nondegenerate λ , the chain complex ECC ∗ ( Y , λ, Γ , J ) is generated as a Z 2 vector space by orbit sets α = { ( α i , m i ) } , which are finite sets for which: α i is an embedded Reeb orbit m i ∈ Z > 0 � i m i [ α i ] = Γ If α i is hyperbolic, m i = 1. The grading ∗ comes from the relative ECH index I ( α, β ), a combination of c 1 (ker λ ), CZ ( α k i ), CZ ( β k j ), and the relative self-intersection. Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Almost complex structures and ∂ ECH A λ -compatible almost complex structure is a complex structure J on T ( R × Y ), for which: J is R -invariant J ξ = ξ , positively with respect to d λ J ( ∂ s ) = R , where s denotes the R coordinate � ∂ ECH α, β � counts currents , disjoint unions of J -holomorphic curves u : ( ˙ Σ , j ) → ( R × Y , J ) , du ◦ j = J ◦ du which are asymptotically cylindrical to orbit sets α and β at ±∞ . For generic J, ECH index one yields somewhere injective. -Hutchings’ Haiku Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Embedded contact homology differential ∂ ECH Theorem (Hutchings-Taubes ’09) ∂ ECH � 2 = 0 , so ( ECC ∗ ( Y , λ, Γ , J ) , ∂ ECH ) is a chain complex. � Theorem (Taubes, Kutluhan-Lee-Taubes, Colin-Ghiggini-Honda) The homology depends only on ( Y , ker λ, Γ) . We denote the homology by ECH ∗ ( Y , ker λ, Γ). Dee squared is zero; obstruction bundle gluing is complicated. -Hutchings-Taubes’ Haiku Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

ECH from H ∗ Theorem (Nelson-Weiler, 90%) Let ( Y , ξ = ker λ ) be a prequantization bundle over (Σ g , ω ) . Then � ECH ∗ ( Y , ξ, Γ) ∼ = Z 2 Λ ∗ H ∗ (Σ g ; Z 2 ) Γ ∈ H 1 ( Y ; Z ) Inspired by the 2011 PhD thesis of Farris. 1 The critical points of a perfect H form a basis for H ∗ (Σ g ; Z 2 ). 1 · · · h m 2 g The generators of ECH are of the form e m − 2 g e m + − h m 1 + where m i = 0 , 1, so correspond to a basis for Λ ∗ H ∗ (Σ g ; Z 2 ). 2 We will prove ∂ ECH only counts cylinders corresponding to Morse flows on Σ g , therefore ∂ ECH ( e m − 1 · · · h m 2 g 2 g e m + − h m 1 + ) is a sum over all ways to apply ∂ Morse to h i or e + . Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Our favorite fibration on S 3 Example ( S 3 , λ ) The ECH of S 3 is the Z 2 -vector space generated by terms e m − − e m + + , where | e − | = 2 , | e + | = 4 . Note that ∗ is not the grading on Λ ∗ H ∗ (Σ g ; Z 2 ) , since | e 2 − | = 6 . fiber: e + The fibers above the critical points of the height function on S 2 represent e ± . We have ∂ ECH = 0 because ∂ Morse = 0. fiber: e − Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Lens spaces L ( k , 1) L ( k , 1) is the total space of the prequantization bundle with Euler number − k on S 2 . Corollary (Nelson-Weiler, 95%) With its prequantization contact structure ξ k ,   if ∗ ∈ 2 Z ≥ 0 Z 2  ECH ∗ ( L ( k , 1) , ξ k , Γ) ∼ =  0 else  for all Γ ∈ H 1 ( L ( k , 1); Z ) . Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

Finer points of the isomorphism Fix a negative Euler class e . For Γ ∈ { 0 , . . . , − e − 1 } , � Λ Γ − ne ( H ∗ (Σ g ; Z 2 )) ECH ∗ ( Y , ξ, Γ) = n ∈ Z ≥ 0 Proposition (Nelson-Weiler) Let α = e m − 1 · · · h m 2 g and let β = e n − 1 · · · h n 2 g − h m 1 2 g e m + − h n 1 2 g e n + + . + j n j and m = ( m − + m + + � i m i ) − N Let N = n − + n + + � . Then − e I ( α, β ) = (2 − 2 g ) m − m 2 e + 2 mN + m + − m − − n + + n − Using this formula, we obtain I ( e N + e , e N − ) = 2 g − 2 + Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles

ECH ∗ ( Y , ξ, 0) for g = 2 , e = − 1 Recall I ( e N + e , e N − ) = 2 g − 2. Set ∗ ( α ) = I ( α, ∅ ). + ∗ = − 2 ∗ = − 1 ∗ = 0 ∗ = 1 ∗ = 2 ∗ = 3 ∗ = 4 Λ 0 ∅ Λ 1 e − h i e + Λ 2 e 2 e 2 e − h i e − e + h i e + + − h i h j Λ 3 e 3 e 2 e 2 e − e 2 − h i − e + e − h i e + · · · + − e − h i h j h i h j h k h i h j e + Λ 4 e 4 · · · − Jo Nelson & Morgan Weiler Embedded Contact Homology of Prequantization Bundles