Poincaré Duality Angles on Riemannian Manifolds with Boundary

Transcription

1 Poincaré Duality Angles on Riemannian Manifolds with Boundary Clayton Shonkwiler Department of Mathematics University of Pennsylvania June 5, 2009

2 Realizing cohomology groups as spaces of differential forms Let M n be a compact Riemannian manifold with non-empty boundary M.

3 Realizing cohomology groups as spaces of differential forms Let M n be a compact Riemannian manifold with non-empty boundary M. H p (M; R) H p (M, M; R)

4 de Rham s Theorem de Rham s Theorem Suppose M n is a compact, oriented, smooth manifold. Then H p (M; R) = C p (M)/E p (M), where C p (M) is the space of closed p-forms on M and E p (M) is the space of exact p-forms.

5 Riemannian metric If M is Riemannian, the metric induces an L 2 inner product on Ω p (M): ω, η := ω η. M

6 Riemannian metric If M is Riemannian, the metric induces an L 2 inner product on Ω p (M): ω, η := ω η. M When M is closed, the orthogonal complement of E p (M) inside C p (M) is H p (M) := {ω Ω p (M) : dω = 0, δω = 0}

7 Riemannian metric If M is Riemannian, the metric induces an L 2 inner product on Ω p (M): ω, η := ω η. M When M is closed, the orthogonal complement of E p (M) inside C p (M) is H p (M) := {ω Ω p (M) : dω = 0, δω = 0} Kodaira called this the space of harmonic p-fields on M.

8 Hodge s Theorem Hodge s Theorem If M n is a closed, oriented, smooth Riemannian manifold, H p (M; R) = H p (M).

9 Hodge Morrey Friedrichs Decomposition Define i : M M to be the natural inclusion.

10 Hodge Morrey Friedrichs Decomposition Define i : M M to be the natural inclusion. The L 2 -orthogonal complement of the exact forms inside the space of closed forms is now: H p N (M) := {ω Ωp (M) : dω = 0, δω = 0, i ω = 0}.

11 Hodge Morrey Friedrichs Decomposition Define i : M M to be the natural inclusion. The L 2 -orthogonal complement of the exact forms inside the space of closed forms is now: H p N (M) := {ω Ωp (M) : dω = 0, δω = 0, i ω = 0}. Then H p (M; R) = H p N (M).

12 Hodge Morrey Friedrichs Decomposition (continued) The relative cohomology appears as H p (M, M; R) = H p D (M).

13 Hodge Morrey Friedrichs Decomposition (continued) The relative cohomology appears as H p (M, M; R) = H p D (M). H p D (M) := {ω Ωp (M) : dω = 0, δω = 0, i ω = 0}.

14 Non-orthogonality The concrete realizations of H p (M; R) and H p (M, M; R) meet only at the origin: H p N (M) Hp D (M) = {0}

15 Non-orthogonality The concrete realizations of H p (M; R) and H p (M, M; R) meet only at the origin: H p N (M) Hp D (M) = {0}...but they are not orthogonal!

16 Non-orthogonality The concrete realizations of H p (M; R) and H p (M, M; R) meet only at the origin: H p N (M) Hp D (M) = {0}...but they are not orthogonal! boundary H p N (M) interior interior boundary H p D (M)

17 Interior and boundary subspaces Interior subspace of H p N (M): ker i where i : H p (M; R) H p ( M; R)

18 Interior and boundary subspaces Interior subspace of H p N (M): ker i where i : H p (M; R) H p ( M; R) E H p N (M) := {ω Hp N (M) : i ω = dϕ, ϕ Ω p 1 ( M)}.

19 Interior and boundary subspaces Interior subspace of H p N (M): ker i where i : H p (M; R) H p ( M; R) E H p N (M) := {ω Hp N (M) : i ω = dϕ, ϕ Ω p 1 ( M)}. Interior subspace of H p D (M): E H n p N (M) = ce H p D (M) = {η H p D (M) : i η = dψ, ψ Ω n p 1 ( M)}.

20 Poincaré duality angles ceh p N (M) H p N (M) E H p N (M) ce H p D (M) EH p D (M) H p D (M) Definition (DeTurck Gluck) The Poincaré duality angles of the Riemannian manifold M are the principal angles between the interior subspaces.

21 What do the Poincaré duality angles tell you? Guess If M is almost closed, the Poincaré duality angles of M should be small.

22 For example... Consider CP 2 with its usual Fubini-Study metric. Let p CP 2. Then define M r := CP 2 B r (p).

23 For example... Consider CP 2 with its usual Fubini-Study metric. Let p CP 2. Then define M r := CP 2 B r (p). π/2 CP 1 p

24 M r is a 3-sphere. Topology of M r

25 Topology of M r M r is a 3-sphere. M r is the D 2 -bundle over CP 1 ( S 2 (1/2)) with Euler characteristic 1.

26 Topology of M r M r is a 3-sphere. M r is the D 2 -bundle over CP 1 ( S 2 (1/2)) with Euler characteristic 1. M r has absolute cohomology in dimensions 0 and 2.

27 Topology of M r M r is a 3-sphere. M r is the D 2 -bundle over CP 1 ( S 2 (1/2)) with Euler characteristic 1. M r has absolute cohomology in dimensions 0 and 2. M r has relative cohomology in dimensions 2 and 4.

28 Topology of M r M r is a 3-sphere. M r is the D 2 -bundle over CP 1 ( S 2 (1/2)) with Euler characteristic 1. M r has absolute cohomology in dimensions 0 and 2. M r has relative cohomology in dimensions 2 and 4. Therefore, M r has a single Poincaré duality angle θ r between H 2 N (M r ) and H 2 D (M r ).

29 Find harmonic 2-fields So the goal is to find closed and co-closed 2-forms on M r which satisfy Neumann and Dirichlet boundary conditions.

30 Find harmonic 2-fields So the goal is to find closed and co-closed 2-forms on M r which satisfy Neumann and Dirichlet boundary conditions. Such 2-forms must be isometry-invariant.

31 Find harmonic 2-fields So the goal is to find closed and co-closed 2-forms on M r which satisfy Neumann and Dirichlet boundary conditions. Such 2-forms must be isometry-invariant. Isom 0 (M r ) = SU(2).

32 Find harmonic 2-fields So the goal is to find closed and co-closed 2-forms on M r which satisfy Neumann and Dirichlet boundary conditions. Such 2-forms must be isometry-invariant. Isom 0 (M r ) = SU(2). Find closed and co-closed SU(2)-invariant forms on M r satisfying Neumann and Dirichlet boundary conditions

33 The Poincaré duality angle for M r cos θ r = 1 sin4 r 1 + sin 4 r.

34 The Poincaré duality angle for M r cos θ r = 1 sin4 r 1 + sin 4 r. As r 0, the Poincaré duality angle θ r 0.

35 The Poincaré duality angle for M r cos θ r = 1 sin4 r 1 + sin 4 r. As r 0, the Poincaré duality angle θ r 0. As r π/2, θ r π/2.

36 The Poincaré duality angle for M r cos θ r = 1 sin4 r 1 + sin 4 r. As r 0, the Poincaré duality angle θ r 0. As r π/2, θ r π/2. Generalizes to CP n B r (p).

37 Poincaré duality angles of Grassmannians Consider N r := G 2 R n ν r (G 1 R n 1 ).

38 Poincaré duality angles of Grassmannians Consider N r := G 2 R n ν r (G 1 R n 1 ). Theorem As r 0, all the Poincaré duality angles of N r go to zero. As r approaches its maximum value of π/2, all the Poincaré duality angles of N r go to π/2.

39 A conjecture Conjecture If M n is a closed Riemannian manifold and N k is a closed submanifold of codimension 2, the Poincaré duality angles of go to zero as r 0. M ν r (N)

40 A question What can you learn about the topology of M from knowledge of M?

41 Electrical Impedance Tomography Induce potentials on the boundary of a region and determine the conductivity inside the region by measuring the current flux through the boundary.

42 Electrical Impedance Tomography Induce potentials on the boundary of a region and determine the conductivity inside the region by measuring the current flux through the boundary.

43 The Voltage-to-Current map Suppose f is a potential on the boundary of a region M R 3.

44 The Voltage-to-Current map Suppose f is a potential on the boundary of a region M R 3. Then f extends to a potential u on M, where u = 0, u M = f.

45 The Voltage-to-Current map Suppose f is a potential on the boundary of a region M R 3. Then f extends to a potential u on M, where u = 0, u M = f. If γ is the conductivity on M, the current flux through M is given by (γ u) ν = γ u ν

46 The Dirichlet-to-Neumann map The map Λ cl : C ( M) C ( M) defined by f u ν is the classical Dirichlet-to-Neumann map.

47 The Dirichlet-to-Neumann map The map Λ cl : C ( M) C ( M) defined by f u ν is the classical Dirichlet-to-Neumann map. Theorem (Lee-Uhlmann) If M n is a compact, analytic Riemannian manifold with boundary, then M is determined up to isometry by Λ cl.

48 Generalization to differential forms Joshi Lionheart and Belishev Sharafutdinov generalized the classical Dirichlet-to-Neumann map to differential forms: Λ : Ω p ( M) Ω n p 1 ( M)

49 Generalization to differential forms Joshi Lionheart and Belishev Sharafutdinov generalized the classical Dirichlet-to-Neumann map to differential forms: Λ : Ω p ( M) Ω n p 1 ( M) Theorem (Belishev Sharafutdinov) The data ( M, Λ) completely determines the cohomology groups of M.

50 Define the Hilbert transform T := dλ 1. Connection to Poincaré duality angles

51 Connection to Poincaré duality angles Define the Hilbert transform T := dλ 1. Theorem If θ 1,..., θ k are the Poincaré duality angles of M in dimension p, then the quantities ( 1) np+n+p cos 2 θ i are the non-zero eigenvalues of an appropriate restriction of T 2.

52 Belishev and Sharafutdinov posed the following question: Can the multiplicative structure of cohomologies be recovered from our data ( M, Λ)? Till now, the authors cannot answer the question. Cup products

53 Belishev and Sharafutdinov posed the following question: Can the multiplicative structure of cohomologies be recovered from our data ( M, Λ)? Till now, the authors cannot answer the question. Cup products Theorem The mixed cup product : H p (M; R) H q (M, M; R) H p+q (M, M; R) is completely determined by the data ( M, Λ) when the relative class is restricted to come from the boundary subspace.

54 Poincaré duality angles for G 4 R 8 ν r (G 3 R 7 )? Other Grassmann manifolds with boundary? Some questions

55 Some questions Poincaré duality angles for G 4 R 8 ν r (G 3 R 7 )? Other Grassmann manifolds with boundary? What is the limiting behavior of the Poincaré duality angles as the manifold closes up?

56 Some questions Poincaré duality angles for G 4 R 8 ν r (G 3 R 7 )? Other Grassmann manifolds with boundary? What is the limiting behavior of the Poincaré duality angles as the manifold closes up? Can the full mixed cup product be recovered from ( M, Λ)? What about other cup products?

57 Some questions Poincaré duality angles for G 4 R 8 ν r (G 3 R 7 )? Other Grassmann manifolds with boundary? What is the limiting behavior of the Poincaré duality angles as the manifold closes up? Can the full mixed cup product be recovered from ( M, Λ)? What about other cup products? Can the L 2 inner product on H p N (M) and Hp D (M) be recovered from ( M, Λ)?

58 Thanks!