A 21st Century Geometry: Contact Geometry

Transcription

1 A 21st Century Geometry: Contact Geometry Bahar Acu University of Southern California California State University Channel Islands September 23, 2015 Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 1 / 12

2 Manifolds Definition A smooth n-manifold is a topological space that looks locally like R n and admits a global differentiable structure. Examples: Trivial example; R n ; Euclidean space S 1 ; compact 1-manifold More generally, S n ; compact n-manifold Torus (doughnut!); closed n-manifold Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 2 / 12

3 Manifolds Definition A smooth n-manifold is a topological space that looks locally like R n and admits a global differentiable structure. Examples: Trivial example; R n ; Euclidean space S 1 ; compact 1-manifold More generally, S n ; compact n-manifold Torus (doughnut!); closed n-manifold Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 2 / 12

4 Tangent Spaces and Differential Forms Definition The tangent space of M n is a vector space at a point p M diffeomorphic to R n. It is denoted by T p M, Definition A 1-form is a linear function: T p M R Differential forms are a coordinate independent approach to calculus. They re great for defining integrals over curves, surfaces, and manifolds! Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 3 / 12

5 Tangent Spaces and Differential Forms Definition The tangent space of M n is a vector space at a point p M diffeomorphic to R n. It is denoted by T p M, Definition A 1-form is a linear function: T p M R Differential forms are a coordinate independent approach to calculus. They re great for defining integrals over curves, surfaces, and manifolds! Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 3 / 12

6 Tangent Spaces and Differential Forms Definition The tangent space of M n is a vector space at a point p M diffeomorphic to R n. It is denoted by T p M, Definition A 1-form is a linear function: T p M R Differential forms are a coordinate independent approach to calculus. They re great for defining integrals over curves, surfaces, and manifolds! Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 3 / 12

7 2-Plane Fields A 2-plane field ξ on M 3 can be written as the kernel of a 1-form. Definition ξ is integrable if at each point p M there is a small open chunk of a surface S in M containing p for which T p S = ξ p. Nice and integrable Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 4 / 12

8 2-Plane Fields A 2-plane field ξ on M 3 can be written as the kernel of a 1-form. Definition ξ is integrable if at each point p M there is a small open chunk of a surface S in M containing p for which T p S = ξ p. Nice and integrable Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 4 / 12

9 2-Plane Fields A 2-plane field ξ on M 3 can be written as the kernel of a 1-form. Definition ξ is integrable if at each point p M there is a small open chunk of a surface S in M containing p for which Tp S = ξp. Nice and integrable Bahar Acu (University of Southern California) Nonintegrable!! A 21st Century Geometry CSUCI Undergraduate Seminar 4 / 12

10 First Contact with Contact Manifolds A 2-plane field ξ is a contact structure if it is nowhere integrable. This is equivalent to saying hyperplanes twist too much to be tangent to hypersurfaces. Rotate a line of planes from + to -. Sweep left-right and up-down. (images from sketches of topology blog) Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 5 / 12

11 First Contact with Contact Manifolds Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 5 / 12

12 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

13 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

14 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

15 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

16 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

17 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

18 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

19 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

20 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

21 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

22 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

23 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

24 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

25 Describing contact structures on manifolds The kernel of a 1-form α on M 2n 1 is a contact structure whenever α (dα)n 1 is a volume form dα ξ is nondegenerate. Here α = dz ydx and ξ = ker α o n, y z + x Span(ξ) = y dα = dy dx = dx dy α dα Also, dα y, y z Bahar Acu (University of Southern California) + x = dz dx dy = dx dy dz. = dx dy y, x = dx y dy x dx = 1. A 21st Century Geometry x dy y CSUCI Undergraduate Seminar 6 / 12

26 Standard Contact Structure on R 3 z y x λ = dz ydx. These planes appear to twist along the y-axis. Another example: All odd dimensional spheres are contact manifolds! Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 7 / 12

27 Standard Contact Structure on R 3 z y x λ = dz ydx. These planes appear to twist along the y-axis. Another example: All odd dimensional spheres are contact manifolds! Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 7 / 12

28 Hopf Fibration Video Fun Fact: Reeb orbits of S 3 are the Hopf fibers of S 3!! The Hopf fibration shows how the three-sphere can be built by a collection of circles arranged like points on a two-sphere. We see specific points on the two-sphere synchronized with the circles (fibers) over them. by Niles Johnson. Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 8 / 12

29 Open Book Decomposition and Giroux Correspondence Definition (Informal) An open book decomposition (or simply an open book) is a decomposition of a closed 3-manifold M into a union of surfaces (necessarily with boundary) and solid tori. Surfaces = Pages, F, of the open book of M Solid Tori = Binding, B, of the open book of M Theorem (Giroux, 2000) Let M be a compact oriented 3-manifold. Then there is a bijection between the set of oriented contact structures on M and the set of open book decompositions of M. That is to say, contact geometry can be studied from an entirely topological viewpoint. Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 9 / 12

30 Open Book Decomposition and Giroux Correspondence Definition (Informal) An open book decomposition (or simply an open book) is a decomposition of a closed 3-manifold M into a union of surfaces (necessarily with boundary) and solid tori. Surfaces = Pages, F, of the open book of M Solid Tori = Binding, B, of the open book of M Theorem (Giroux, 2000) Let M be a compact oriented 3-manifold. Then there is a bijection between the set of oriented contact structures on M and the set of open book decompositions of M. That is to say, contact geometry can be studied from an entirely topological viewpoint. Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 9 / 12

31 Dehn Twists as monodromy (Photo courtesy: Jonny Evans) Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 10 / 12

32 Factorization of Monodromy is Possible! Theorem (Acu-Avdek 14) Given a homogeneous polynomial f C[z 0,..., z n ] of degree k with an isolated singularity at 0, a 2n-dimensional Weinstein domain (W, dβ) where W = {f (z 0,..., z n ) = 0} B 2n+2 Then the contact manifold W has an open book OB(F, Φ ) such that a fibered Dehn twist Φ along W can be expressed as a product of k(k 1) n right-handed Dehn twists Φ 1... Φ k(k 1) n(up to symplectic isotopy). Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 11 / 12

33 Thanks for listening! Special thanks to Jo Nelson for sharing contact plane distribution images with me! Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 12 / 12