# AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY

Size: px
Start display at page:

Transcription

1 1 AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY Andrew Ranicki and Daniele Sepe (Edinburgh) aar Maslov index seminar, 9 November 2009

2 The 1-dimensional Lagrangians 2 Definition (i) Let R 2 have the symplectic form [, ] : R 2 R 2 R ; ((x 1, y 1 ), (x 2, y 2 )) x 1 y 2 x 2 y 1. (ii) A subspace L R 2 is Lagrangian of (R 2, [, ]) if L = L = {x R 2 [x, y] = 0 for all y L}. Proposition A subspace L R 2 is a Lagrangian of (R 2, [, ]) if and only if L is 1-dimensional, Definition (i) The 1-dimensional Lagrangian Grassmannian Λ(1) is the space of Lagrangians L (R 2, [, ]), i.e. the Grassmannian of 1-dimensional subspaces L R 2. (ii) For θ R let L(θ) = {(rcos θ, rsin θ) r R} Λ(1) be the Lagrangian with gradient tan θ.

3 The topology of Λ(1) 3 Proposition The square function det 2 and the square root function : Λ(1) S 1 ; L(θ) e 2iθ ω : S 1 Λ(1) = R P 1 ; e 2iθ L(θ) are inverse diffeomorphisms, and π 1 (Λ(1)) = π 1 (S 1 ) = Z. Proof Every Lagrangian L in (R 2, [, ]) is of the type L(θ), and L(θ) = L(θ ) if and only if θ θ = kπ for some k Z. Thus there is a unique θ [0, π) such that L = L(θ). The loop ω : S 1 Λ(1) represents the generator ω = 1 π 1 (Λ(1)) = Z.

4 4 The Maslov index of a 1-dimensional Lagrangian Definition The Maslov index of a Lagrangian L = L(θ) in (R 2, [, ]) is 1 2θ if 0 < θ < π τ(l) = π 0 if θ = 0 The Maslov index for Lagrangians in (R 2, [, ]) is reduced to the n special case n = 1 by the diagonalization of unitary matrices. The formula for τ(l) has featured in many guises besides the Maslov index (e.g. as assorted η-, γ-, ρ-invariants and an L 2 -signature) in the papers of Arnold (1967), Neumann (1978), Atiyah (1987), Cappell, Lee and Miller (1994), Bunke (1995), Nemethi (1995), Cochran, Orr and Teichner (2003),... See aar/maslov.htm for detailed references.

5 5 The Maslov index of a pair of 1-dimensional Lagrangians Note The function τ : Λ(1) R is not continuous. Examples τ(l(0)) = τ(l( π 2 )) = 0, τ(l(π 4 )) = 1 2 R with L(0) = R 0, L( π 2 ) = 0 R, L(π ) = {(x, x) x R}. 4 Definition The Maslov index of a pair of Lagrangians (L 1, L 2 ) = (L(θ 1 ), L(θ 2 )) in (R 2, [, ]) is 1 2(θ 2 θ 1 ) if 0 θ 1 < θ 2 < π τ(l 1, L 2 ) = τ(l 2, L 1 ) = π 0 if θ 1 = θ 2. Examples τ(l) = τ(r 0, L), τ(l, L) = 0.

6 6 The Maslov index of a triple of 1-dimensional Lagrangians Definition The Maslov index of a triple of Lagrangians in (R 2, [, ]) is (L 1, L 2, L 3 ) = (L(θ 1 ), L(θ 2 ), L(θ 3 )) τ(l 1, L 2, L 3 ) = τ(l 1, L 2 ) + τ(l 2, L 3 ) + τ(l 3, L 1 ) { 1, 0, 1} R. Example If 0 θ 1 < θ 2 < θ 3 < π then τ(l 1, L 2, L 3 ) = 1 Z. Example The Wall computation of the signature of C P 2 is given in terms of the Maslov index as σ(c P 2 ) = τ(l(0), L(π/4), L(π/2)) = τ(l(0), L(π/4)) + τ(l(π/4), L(π/2)) + τ(l(π/2), L(0)) = = 1 Z R.

7 The Maslov index and the degree I. 7 A pair of 1-dimensional Lagrangians (L 1, L 2 ) = (L(θ 1 ), L(θ 2 )) determines a path in Λ(1) from L 1 to L 2 ω 12 : I Λ(1) ; t L((1 t)θ 1 + tθ 2 ). For any L = L(θ) Λ(1)\{L 1, L 2 } (ω 12 ) 1 (L) = {t [0, 1] L((1 t)θ 1 + tθ 2 ) = L} = {t [0, 1] (1 t)θ 1 + tθ 2 = θ} { } θ θ1 if 0 < θ θ 1 < 1 = θ 2 θ 1 θ 2 θ 1 otherwise. The degree of a loop ω : S 1 Λ(1) = S 1 is the number of elements in ω 1 (L) for a generic L Λ(1). In the geometric applications the Maslov index counts the number of intersections of a curve in a Lagrangian manifold with the codimension 1 cycle of singular points.

8 8 The Maslov index and the degree II. Proposition A triple of Lagrangians (L 1, L 2, L 3 ) determines a loop in Λ(1) ω 123 = ω 12 ω 23 ω 31 : S 1 Λ(1) with homotopy class the Maslov index of the triple ω 123 = τ(l 1, L 2, L 3 ) { 1, 0, 1} π 1 (Λ(1)) = Z. Proof It is sufficient to consider the special case with 0 θ 1 < θ 2 < θ 3 < π, so that (L 1, L 2, L 3 ) = (L(θ 1 ), L(θ 2 ), L(θ 3 )) det 2 ω 123 = 1 : S 1 S 1, degree(det 2 ω 123 ) = 1 = τ(l 1, L 2, L 3 ) Z

9 9 The Euclidean structure on R 2n The phase space is the 2n-dimensional Euclidean space R 2n, with preferred basis {p 1, p 2,..., p n, q 1, q 2,..., q n }. The 2n-dimensional phase space carries 4 additional structures. Definition The Euclidean structure on R 2n is the positive definite symmetric form over R (, ) : R 2n R 2n R ; (v, v ) n x j x j + n y k y k, j=1 k=1 (v = n x j p j + n y k q k, v = n x j p j + n y k q k R 2n ). j=1 k=1 The automorphism group of (R 2n, (, )) is the orthogonal group O(2n) of invertible 2n 2n matrices A = (a jk ) (a jk R) such that A A = I 2n with A = (a kj ) the transpose. j=1 k=1

10 10 The complex structure on R 2n Definition The complex structure on R 2n is the linear map ı : R 2n R 2n ; n n x j p j + y k q k j=1 k=1 n n x j p j y k q k j=1 k=1 such that ı 2 = 1. Use ı to regard R 2n as an n-dimensional complex vector space, with an isomorphism R 2n C n ; v (x 1 + iy 1, x 2 + iy 2,..., x n + iy n ). The automorphism group of (R 2n, ı) = C n is the complex general linear group GL(n, C) of invertible n n matrices (a jk ) (a jk C).

11 11 The symplectic structure on R 2n Definition The symplectic structure on R 2n is the symplectic form [, ] : R 2n R 2n R ; (v, v ) [v, v ] = (ıv, v ) = [v, v] = n (x j y j x j y j ). The automorphism group of (R 2n, [, ]) is the symplectic group Sp(n) of invertible 2n 2n matrices A = (a jk ) (a jk R) such that ( ) A 0 In A = I n 0 ( 0 ) In I n 0 j=1.

12 The n-dimensional Lagrangians Definition Given a finite-dimensional real vector space V with a nonsingular symplectic form [, ] : V V R let Λ(V ) be the set of Lagrangian subspaces L V, with L = L = {x V [x, y] = 0 R for all y L}. Terminology Λ(R 2n ) = Λ(n). The real and imaginary Lagrangians n n R n = { x j p j x j R n }, ır n = { y k q k y k R n } Λ(n) j=1 are complementary, with R 2n = R n ır n. Definition The graph of a symmetric form (R n, φ) is the Lagrangian n n n Γ (R n,φ) = {(x, φ(x)) x = x j p j, φ(x) = φ jk x j q k } Λ(n) j=1 k=1 j=1 k=1 complementary to ır n. Proposition Every Lagrangian complementary to ır n is a graph. 12

13 13 The hermitian structure on R 2n Definition The hermitian inner product on R 2n is defined by, : R 2n R 2n C ; (v, v ) v, v = (v, v ) + i[v, v ] = n j=1 (x j + iy j )(x j iy j ). or equivalently by, : C n C n C ; (z, z ) z, z = n z j z j. j=1 The automorphism group of (C n,, ) is the unitary group U(n) of invertible n n matrices A = (a jk ) (a jk C) such that AA = I n, with A = (a kj ) the conjugate transpose.

14 14 The general linear, orthogonal and unitary groups Proposition (Arnold, 1967) (i) The automorphism groups of R 2n with respect to the various structures are related by O(2n) GL(n, C) = GL(n, C) Sp(n) = Sp(n) O(2n) = U(n). (ii) The determinant map det : U(n) S 1 is the projection of a fibre bundle SU(n) U(n) S 1. (iii) Every A U(n) sends the standard Lagrangian ır n of (R 2n, [, ]) to a Lagrangian A(ıR n ). The unitary matrix A = (a jk ) is such that A(ıR n ) = ır n if and only if each a jk R C, with O(n) = {A U(n) A(ıR n ) = ır n } U(n).

15 15 The Lagrangian Grassmannian Λ(n) I. Λ(n) is the space of all Lagrangians L (R 2n, [, ]). Proposition (Arnold, 1967) The function is a diffeomorphism. U(n)/O(n) Λ(n) ; A A(ıR n ) Λ(n) is a compact manifold of dimension dim Λ(n) = dim U(n) dim O(n) = n 2 n(n 1) 2 The graphs {Γ (R n,φ) φ = φ M n (R)} Λ(n) define a chart at R n Λ(n). Example (Arnold and Givental, 1985) = n(n + 1) 2. Λ(2) 3 = {[x, y, z, u, v] R P 4 x 2 + y 2 + z 2 = u 2 + v 2 } = S 2 S 1 /{(x, y) ( x, y)}.

16 The Lagrangian Grassmannian Λ(n) II. 16 In view of the fibration Λ(n) = U(n)/O(n) BO(n) BU(n) Λ(n) classifies real n-plane bundles β with a trivialisation δβ : C β = ɛ n of the complex n-plane bundle C β. The canonical real n-plane bundle η over Λ(n) is The complex n-plane bundle C η E(η) = {(L, l) L Λ(n), l L}. E(C η) = {(L, l C ) L Λ(n), l C C R L} is equipped with the canonical trivialisation δη : C η = ɛ n defined by δη : E(C η) = E(ɛ n ) = Λ(n) C n ; (L, l C ) (L, (p, q)) if l C = (p, q) C R L = L il = C n.

17 The fundamental group π 1 (Λ(n)) I. 17 Theorem (Arnold, 1967) The square of the determinant function det 2 : Λ(n) S 1 ; L = A(ıR n ) det(a) 2 induces an isomorphism det 2 = : π 1 (Λ(n)) π 1 (S 1 ) = Z. Proof By the homotopy exact sequence of the commutative diagram of fibre bundles det SO(n) O(n) O(1) = S 0 SU(n) SΛ(n) U(n) Λ(n) det U(1) = S 1 z z 2 det 2 Λ(1) = S 1

18 18 Corollary 1 The function The fundamental group π 1 (Λ(n)) = Z II. π 1 (Λ(n)) Z ; (ω : S 1 Λ(n)) degree( S 1 ω Λ(n) det2 S 1 ) is an isomorphism. Corollary 2 The cohomology class α H 1 (Λ(n)) = Hom(π 1 (Λ(n)), Z) characterized by α(ω) = degree( S 1 ω Λ(n) det2 S 1 ) Z is a generator α H 1 (Λ(n)) = Z.

19 π 1 (Λ(1)) = Z in terms of line bundles Example The universal real line bundle η : S 1 BO(1) = R P is the infinite Möbius band over S 1 E(η) = R [0, π]/{(x, 0) ( x, π)} S 1 = [0, π]/(0 π). The complexification C η : S 1 BU(1) = C P is the trivial complex line bundle over S 1 E(C η) = C [0, π]/{(z, 0) ( z, π)} S 1 = [0, π]/(0 π). The canonical trivialisation δη : C η = ɛ is defined by = δη : E(C η) E(ɛ) = C [0, π]/{(z, 0) (z, π)} = C S 1 ; [z, θ] [ze iθ, θ]. The canonical pair (δη, η) represents (δη, η) = 1 π 1 (Λ(1)) = π 1 (U(1)/O(1)) = Z, corresponding to the loop ω : S 1 Λ(1); e 2iθ L(θ). 19

20 The Maslov index for n-dimensional Lagrangians Unitary matrices can be diagonalized. For every A U(n) there exists B U(n) such that BAB 1 = D(e iθ 1, e iθ 2,..., e iθn ) is the diagonal matrix, with e iθ j S 1 the eigenvalues, i.e. the roots of the characteristic polynomial n ch z (A) = det(zi n A) = (z e iθ j ) C[z]. j=1 Definition The Maslov index of L Λ(n) is n τ(l) = (1 2θ j /π) R j=1 with θ 1, θ 2,..., θ n [0, π) such that ±e iθ 1, ±e iθ 2,..., ±e iθn are the eigenvalues of any A U(n) such that A(ıR n ) = L. There are similar definitions of τ(l, L ) R and τ(l, L, L ) Z. 20

21 21 The Maslov cycle Corollary 2 above shows that the Maslov index α H 1 (Λ(n)) is the pullback along det 2 : Λ(n) S 1 of the standard form dθ on S 1. Definition The Poincaré dual of α is the Maslov cycle m = {L Λ(n) L ır n {0}}. In his 1967 paper Arnold constructs this cycle following ideas that generalise the standard constructions on Schubert varieties in the Grassmannians of linear subspaces of Euclidean spaces, and proves that [m] H (n+2)(n 1)/2 (Λ(n)) is the Poincaré dual of α H 1 (Λ(n)).

22 Lagrangian submanifolds of R 2n I. 22 An n-dimensional submanifold M n R 2n is Lagrangian if for each x M the tangent n-plane T x (M) T x (R 2n ) = R 2n is a Lagrangian subspace of (R 2n, [, ]). The tangent bundle T M : M BO(n) is the pullback of the canonical real n-plane bundle over Λ(n) E(η) = {(λ, l) λ Λ(n), l λ} along the classifying map ζ : M Λ(n); x T x (M), with T M : M ζ η Λ(n) BO(n). The complex n-plane bundle C T M = T C n M has a canonical trivialisation. Definition The Maslov index of a Lagrangian submanifold M n R 2n is the pullback of the generator α H 1 (Λ(n)) τ(m) = ζ (α) H 1 (M).

23 Lagrangian submanifolds of R 2n II. Given a Lagrangian submanifold M n R 2n let π = proj : M n R n ; (p, q) p be the restriction of the projection R 2n = R n ır n R n. The differentials of π are the differentiable maps dπ x : T x (M) T π(x) (R n ) = R n ; l p if l = (p, q) T x (M) R 2n with ker(dπ x ) = T x (M) ır n ır n. If π is a local diffeomorphism then each dπ x is an isomorphism of real vector spaces, with kernel T x (M) ır n = {0}. Definition The Maslov cycle of a Lagrangian submanifold M R 2n is m = {x M T x (M) ır n {0}}. Theorem (Arnold, 1967) Generically, m M is a union of open submanifolds of codimension 1. The homology class [m] H n 1 (M) is the Poincaré dual of the Maslov index class τ(m) H 1 (M), measuring the failure of π : M R n to be a local diffeomorphism. 23

24 24 Symplectic manifolds Definition A manifold N is symplectic if it admits a 2-form ω which is closed and non-degenerate. Examples S 2, R 2n = C n = T R n ; the cotangent bundle T B of any manifold B with a canonical symplectic form. Definition A submanifold M N of a symplectic manifold is Lagrangian if ω M = 0 and each T x M T x N (x M) is a Lagrangian subspace. Remark A symplectic manifold N is necessarily even dimensional (because of the non-degeneracy of ω) and the dimension of a Lagrangian submanifold M is necessarily half the dimension of the ambient manifold. Examples If N is two dimensional, any one dimensional submanifold is Lagrangian (why?). If N = T B for some manifold B with the canonical symplectic form, then each cotangent space T b B is a Lagrangian submanifold.

25 The Λ(n)-affine structure of Λ(V ) Let V be a 2n-dimensional real vector space with nonsingular symplectic form [, ] : V V R and a complex structure ı : V V such that V V R; (v, w) [ıv, w] is an inner product. Let U(V ) be the group of automorphisms A : (V, [, ], ı) (V, [, ], ı). For L Λ(V ) let O(L) = {A U(V ) A(L) = L}. Proposition (Guillemin and Sternberg) (i) The function U(V )/O(L) Λ(V ) ; A A(L) is a diffeomorphism. (ii) There exists an isomorphism f L : (R 2n, [, ], ı) = (V, [, ], ı) such that f L (ır n ) = L. (iii) For any f L as in (ii) the diffeomorphism f L : Λ(n) = U(n)/O(n) Λ(V ) = U(V )/O(L) ; λ f L (λ) only depends on L, and not on the choice of f L. Thus Λ(V ) has a Λ(n)-affine structure: for any L 1, L 2 Λ(V ) there is defined a difference element (L 1, L 2 ) Λ(n), with (L 1, L 1 ) = ır n R 2n 25

26 26 An application of the Maslov index I. For any n-dimensional manifold B the tangent bundle of the cotangent bundle E = T(T B) T B has each fibre a 2n-dimensional symplectic vector space. The symplectic form is given in canonical (or Darboux) coordinates as ω = dp dq (q B, p T qb). The bundle of Lagrangian planes Π : Λ(E) T B is the fibre bundle with fibres the affine Λ(n)-sets Π 1 (x) = Λ(T x T B) of Lagrangians in the 2n-dimensional symplectic vector space T x T B.

27 27 An application of the Maslov index II. The projection π : T B B induces the vertical subbundle V = ker dπ T(T B), whose fibres are Lagrangian subspaces. The map s : T B Λ(E) ; x = (p, q) V x = ker(dπ x : T x (T B) T π(x) (B)) is a section of the bundle of Lagrangian planes. For any other section r : T B Λ(E) use the Λ(n)-affine structures of the fibres Π 1 (x) to define a difference map ζ = (r, s) : T B Λ(n) ; x (r(x), s(x))

28 28 An application of the Maslov index III. Let M be a Lagrangian submanifold of T B, dim M = dim B = n. If π M : M B is a local diffeomorphism then each d(π M ) x : T x (M) T π(x) (B) is an isomorphism of vector bundles, with kernel T x (M) V x = {0}. The Maslov index measures the failure of π M : M B to be a local diffeomorphism, in general. We let E M = TM T(T B), Λ(E M ) = Π 1 (M) Λ(E) so that Π : Λ(E M ) M is a fibre bundle with the fibre over x M the Λ(n)-affine set (Π ) 1 (x) = Λ(T x M).

29 An application of the Maslov index IV. 29 The maps r, s M : M Λ(E M ) defined by r(x) = T x M, s M (x) = V x T x (T B) are sections of Π. The difference map ζ = (r, s M ) : M Λ(n) classifies the real n-plane bundle over M with complex trivialisation in which the fibre over x M is the Lagrangian (T x M, V x ) Λ(n). Definition The Maslov index class is the pullback along ζ of the generator α = 1 H 1 (Λ(n)) = Z α M = ζ α H 1 (M). Proposition The homology class [m] H n 1 (M) of the Maslov cycle m = {x M T x M V x {0}}, is the Poincaré dual of the Maslov index class α M H 1 (M). If the class [m] = α M H n 1 (M) = H 1 (M) is non-zero then the projection π M : M B fails to be a local diffeomorphism.

30 30 An application of the Maslov index V. (Butler, 2007) This formulation of the Maslov index can be used to prove that if Σ is an n-dimensional manifold which is topologically T n, and which admits a geometrically semi-simple convex Hamiltonian, then Σ has the standard differentiable structure.

31 31 References V.I.Arnold, On a characteristic class entering into conditions of quantisation, Functional Analysis and Its Applications 1, 1-14 (1967) V.I.Arnold and A.B.Givental, Symplectic geometry, in Dynamical Systems IV, Encycl. of Math. Sciences 4, Springer, (1990) L.Butler, The Maslov cocycle, smooth structures and real-analytic complete integrability, to appear in Amer. J. Maths. V.Guillemin and S.Sternberg, Symplectic geometry, Chapter IV of Geometric Optics, Math. Surv. and Mon. 14, AMS, (1990)

### Drawing: Prof. Karl Heinrich Hofmann

Drawing: Prof. Karl Heinrich Hofmann 1 2 Sylvester s Law of Inertia A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form

### THE EULER CHARACTERISTIC OF A LIE GROUP

THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth

### SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

### L6: Almost complex structures

L6: Almost complex structures To study general symplectic manifolds, rather than Kähler manifolds, it is helpful to extract the homotopy-theoretic essence of having a complex structure. An almost complex

### SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces

### EXOTIC SMOOTH STRUCTURES ON TOPOLOGICAL FIBRE BUNDLES B

EXOTIC SMOOTH STRUCTURES ON TOPOLOGICAL FIBRE BUNDLES B SEBASTIAN GOETTE, KIYOSHI IGUSA, AND BRUCE WILLIAMS Abstract. When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic,

### Problem set 2. Math 212b February 8, 2001 due Feb. 27

Problem set 2 Math 212b February 8, 2001 due Feb. 27 Contents 1 The L 2 Euler operator 1 2 Symplectic vector spaces. 2 2.1 Special kinds of subspaces....................... 3 2.2 Normal forms..............................

### 1 Smooth manifolds and Lie groups

An undergraduate approach to Lie theory Slide 1 Andrew Baker, Glasgow Glasgow, 12/11/1999 1 Smooth manifolds and Lie groups A continuous g : V 1 V 2 with V k R m k open is called smooth if it is infinitely

### LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.

### arxiv: v1 [math.sg] 6 Nov 2015

A CHIANG-TYPE LAGRANGIAN IN CP ANA CANNAS DA SILVA Abstract. We analyse a simple Chiang-type lagrangian in CP which is topologically an RP but exhibits a distinguishing behaviour under reduction by one

### LECTURE 1: LINEAR SYMPLECTIC GEOMETRY

LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Contents 1. Linear symplectic structure 3 2. Distinguished subspaces 5 3. Linear complex structure 7 4. The symplectic group 10 *********************************************************************************

### GEOMETRIC QUANTIZATION

GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical

### A little taste of symplectic geometry

A little taste of symplectic geometry Timothy Goldberg Thursday, October 4, 2007 Olivetti Club Talk Cornell University 1 2 What is symplectic geometry? Symplectic geometry is the study of the geometry

### LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory

### Complex structures on 4-manifolds with symplectic 2-torus actions

Complex structures on 4-manifolds with symplectic 2-torus actions J.J. Duistermaat and A. Pelayo Abstract We apply the general theory for symplectic torus actions with symplectic or coisotropic orbits

### Morse theory and stable pairs

Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction

### A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds

arxiv:math/0312251v1 [math.dg] 12 Dec 2003 A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds Haibao Duan Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, dhb@math.ac.cn

### The Maslov Index. Seminar Series. Edinburgh, 2009/10

The Maslov Index Seminar Series Edinburgh, 2009/10 Contents 1 T. Thomas: The Maslov Index 3 1.1 Introduction: The landscape of the Maslov index............... 3 1.1.1 Symplectic vs. projective geometry...................

### 1. Classifying Spaces. Classifying Spaces

Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.

### CHARACTERISTIC CLASSES

1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact

### Symplectic Geometry. Spring School, June 7 14, 2004, Utrecht

Symplectic Geometry Spring School, June 7 14, 2004, Utrecht J.J. Duistermaat Department of Mathematics, Utrecht University, Postbus 80.010, 3508 TA Utrecht, The Netherlands. E-mail: duis@math.uu.nl Contents

### The Atiyah bundle and connections on a principal bundle

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 3, June 2010, pp. 299 316. Indian Academy of Sciences The Atiyah bundle and connections on a principal bundle INDRANIL BISWAS School of Mathematics, Tata

### NONCOMMUTATIVE LOCALIZATION IN ALGEBRA AND TOPOLOGY Andrew Ranicki (Edinburgh) aar. Heidelberg, 17th December, 2008

1 NONCOMMUTATIVE LOCALIZATION IN ALGEBRA AND TOPOLOGY Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ aar Heidelberg, 17th December, 2008 Noncommutative localization Localizations of noncommutative

### DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM

DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM ARIEL HAFFTKA 1. Introduction In this paper we approach the topology of smooth manifolds using differential tools, as opposed to algebraic ones such

### LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in

### Math 215B: Solutions 1

Math 15B: Solutions 1 Due Thursday, January 18, 018 (1) Let π : X X be a covering space. Let Φ be a smooth structure on X. Prove that there is a smooth structure Φ on X so that π : ( X, Φ) (X, Φ) is an

### REPRESENTATION THEORY WEEK 7

REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable

### A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM CONTENTS

A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM TIMOTHY E. GOLDBERG ABSTRACT. This is a handout for a talk given at Bard College on Tuesday, 1 May 2007 by the author. It gives careful versions

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### ERRATA FOR INTRODUCTION TO SYMPLECTIC TOPOLOGY

ERRATA FOR INTRODUCTION TO SYMPLECTIC TOPOLOGY DUSA MCDUFF AND DIETMAR A. SALAMON Abstract. These notes correct a few typos and errors in Introduction to Symplectic Topology (2nd edition, OUP 1998, reprinted

### Poisson geometry of b-manifolds. Eva Miranda

Poisson geometry of b-manifolds Eva Miranda UPC-Barcelona Rio de Janeiro, July 26, 2010 Eva Miranda (UPC) Poisson 2010 July 26, 2010 1 / 45 Outline 1 Motivation 2 Classification of b-poisson manifolds

### Cohomology of the Mumford Quotient

Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten

### Patrick Iglesias-Zemmour

Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries

### Hamiltonian Toric Manifolds

Hamiltonian Toric Manifolds JWR (following Guillemin) August 26, 2001 1 Notation Throughout T is a torus, T C is its complexification, V = L(T ) is its Lie algebra, and Λ V is the kernel of the exponential

### Complex manifolds, Kahler metrics, differential and harmonic forms

Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on

### THE ASYMPTOTIC MASLOV INDEX AND ITS APPLICATIONS

THE ASYMPTOTIC MASLOV INDEX AND ITS APPLICATIONS GONZALO CONTRERAS, JEAN-MARC GAMBAUDO, RENATO ITURRIAGA, AND GABRIEL P. PATERNAIN Abstract. Let N be a 2n-dimensional manifold equipped with a symplectic

### LECTURE 4: SYMPLECTIC GROUP ACTIONS

LECTURE 4: SYMPLECTIC GROUP ACTIONS WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic circle actions We set S 1 = R/2πZ throughout. Let (M, ω) be a symplectic manifold. A symplectic S 1 -action on (M, ω) is

### LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT FALL 2006 PRINCETON UNIVERSITY

LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT 204 - FALL 2006 PRINCETON UNIVERSITY ALFONSO SORRENTINO 1 Adjoint of a linear operator Note: In these notes, V will denote a n-dimensional euclidean vector

### Geometric control and dynamical systems

Université de Nice - Sophia Antipolis & Institut Universitaire de France 9th AIMS International Conference on Dynamical Systems, Differential Equations and Applications Control of an inverted pendulum

### Stable moduli spaces of high dimensional manifolds

of high dimensional manifolds University of Copenhagen August 23, 2012 joint work with Søren Galatius Moduli spaces of manifolds We define W g := # g S n S n, a closed 2n-dimensional smooth manifold. When

### NOTES ON FIBER BUNDLES

NOTES ON FIBER BUNDLES DANNY CALEGARI Abstract. These are notes on fiber bundles and principal bundles, especially over CW complexes and spaces homotopy equivalent to them. They are meant to supplement

### EXACT BRAIDS AND OCTAGONS

1 EXACT BRAIDS AND OCTAGONS Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ aar Lewisfest, Dublin, 23 July 2009 Organized by Eva Bayer-Fluckiger, David Lewis and Andrew Ranicki. 2 3 Exact braids

### Vector fields in the presence of a contact structure

Vector fields in the presence of a contact structure Valentin Ovsienko To cite this version: Valentin Ovsienko. Vector fields in the presence of a contact structure. Preprint ICJ. 10 pages. 2005.

### THE NEWLANDER-NIRENBERG THEOREM. GL. The frame bundle F GL is given by x M Fx

THE NEWLANDER-NIRENBERG THEOREM BEN MCMILLAN Abstract. For any kind of geometry on smooth manifolds (Riemannian, Complex, foliation,...) it is of fundamental importance to be able to determine when two

### Math 231b Lecture 16. G. Quick

Math 231b Lecture 16 G. Quick 16. Lecture 16: Chern classes for complex vector bundles 16.1. Orientations. From now on we will shift our focus to complex vector bundles. Much of the theory for real vector

### Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures.

Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures. Andrey Kustarev joint work with V. M. Buchstaber, Steklov Mathematical Institute

### On a generalized Sturm theorem

On a generalized Sturm theorem Alessandro Portaluri May 21, 29 Abstract Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive

### LECTURE 2: SYMPLECTIC VECTOR BUNDLES

LECTURE 2: SYMPLECTIC VECTOR BUNDLES WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic Vector Spaces Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over

### Geometry and Dynamics of singular symplectic manifolds. Session 9: Some applications of the path method in b-symplectic geometry

Geometry and Dynamics of singular symplectic manifolds Session 9: Some applications of the path method in b-symplectic geometry Eva Miranda (UPC-CEREMADE-IMCCE-IMJ) Fondation Sciences Mathématiques de

### 12 Geometric quantization

12 Geometric quantization 12.1 Remarks on quantization and representation theory Definition 12.1 Let M be a symplectic manifold. A prequantum line bundle with connection on M is a line bundle L M equipped

### Notes on the geometry of Lagrangian torus fibrations

Notes on the geometry of Lagrangian torus fibrations U. Bruzzo International School for Advanced Studies, Trieste bruzzo@sissa.it 1 Introduction These notes have developed from the text of a talk 1 where

### LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS

LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS Contents 1. Almost complex manifolds 1. Complex manifolds 5 3. Kähler manifolds 9 4. Dolbeault cohomology 11 1. Almost complex manifolds Almost complex structures.

### (1) Let π Ui : U i R k U i be the natural projection. Then π π 1 (U i ) = π i τ i. In other words, we have the following commutative diagram: U i R k

1. Vector Bundles Convention: All manifolds here are Hausdorff and paracompact. To make our life easier, we will assume that all topological spaces are homeomorphic to CW complexes unless stated otherwise.

### LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds

### HAMILTONIAN ACTIONS IN GENERALIZED COMPLEX GEOMETRY

HAMILTONIAN ACTIONS IN GENERALIZED COMPLEX GEOMETRY TIMOTHY E. GOLDBERG These are notes for a talk given in the Lie Groups Seminar at Cornell University on Friday, September 25, 2009. In retrospect, perhaps

### Stable generalized complex structures

Stable generalized complex structures Gil R. Cavalcanti Marco Gualtieri arxiv:1503.06357v1 [math.dg] 21 Mar 2015 A stable generalized complex structure is one that is generically symplectic but degenerates

### COMPUTABILITY AND THE GROWTH RATE OF SYMPLECTIC HOMOLOGY

COMPUTABILITY AND THE GROWTH RATE OF SYMPLECTIC HOMOLOGY MARK MCLEAN arxiv:1109.4466v1 [math.sg] 21 Sep 2011 Abstract. For each n greater than 7 we explicitly construct a sequence of Stein manifolds diffeomorphic

### Topological K-theory

Topological K-theory Robert Hines December 15, 2016 The idea of topological K-theory is that spaces can be distinguished by the vector bundles they support. Below we present the basic ideas and definitions

### Twisted Poisson manifolds and their almost symplectically complete isotropic realizations

Twisted Poisson manifolds and their almost symplectically complete isotropic realizations Chi-Kwong Fok National Center for Theoretical Sciences Math Division National Tsing Hua University (Joint work

### Master of Science in Advanced Mathematics and Mathematical Engineering

Master of Science in Advanced Mathematics and Mathematical Engineering Title: Constraint algorithm for singular k-cosymplectic field theories Author: Xavier Rivas Guijarro Advisor: Francesc Xavier Gràcia

### Legendrian Approximations

U.U.D.M. Project Report 2014:22 Legendrian Approximations Dan Strängberg Examensarbete i matematik, 30 hp Handledare och examinator: Tobias Ekholm Juni 2014 Department of Mathematics Uppsala University

### The Classical Groups and Domains

September 22, 200 The Classical Groups and Domains Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ The complex unit disk D { C : < has four families of generaliations to bounded open

### MATH 583A REVIEW SESSION #1

MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),

### TRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE. Luis Guijarro and Gerard Walschap

TRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE Luis Guijarro and Gerard Walschap Abstract. In this note, we examine the relationship between the twisting of a vector bundle ξ over a manifold

### Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1

Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines

### CLASSICAL GROUPS DAVID VOGAN

CLASSICAL GROUPS DAVID VOGAN 1. Orthogonal groups These notes are about classical groups. That term is used in various ways by various people; I ll try to say a little about that as I go along. Basically

### Deformations of coisotropic submanifolds in symplectic geometry

Deformations of coisotropic submanifolds in symplectic geometry Marco Zambon IAP annual meeting 2015 Symplectic manifolds Definition Let M be a manifold. A symplectic form is a two-form ω Ω 2 (M) which

### Lecture on Equivariant Cohomology

Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove

### The Maximum Likelihood Degree of Mixtures of Independence Models

SIAM J APPL ALGEBRA GEOMETRY Vol 1, pp 484 506 c 2017 Society for Industrial and Applied Mathematics The Maximum Likelihood Degree of Mixtures of Independence Models Jose Israel Rodriguez and Botong Wang

### The Strominger Yau Zaslow conjecture

The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex

### LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9

### Geometry of the momentum map

Geometry of the momentum map Gert Heckman Peter Hochs February 18, 2005 Contents 1 The momentum map 1 1.1 Definition of the momentum map.................. 2 1.2 Examples of Hamiltonian actions..................

### HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

### A geometric solution of the Kervaire Invariant One problem

A geometric solution of the Kervaire Invariant One problem Petr M. Akhmet ev 19 May 2009 Let f : M n 1 R n, n = 4k + 2, n 2 be a smooth generic immersion of a closed manifold of codimension 1. Let g :

### Math 550 / David Dumas / Fall Problems

Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;

### Metrics and Holonomy

Metrics and Holonomy Jonathan Herman The goal of this paper is to understand the following definitions of Kähler and Calabi-Yau manifolds: Definition. A Riemannian manifold is Kähler if and only if it

### LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

### Practice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009.

Practice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009. Solutions (1) Let Γ be a discrete group acting on a manifold M. (a) Define what it means for Γ to act freely. Solution: Γ acts

### Lagrangian and Legendrian Singularities

Lagrangian and Legendrian Singularities V.V.Goryunov and V.M.Zakalyukin These are notes of the mini-courses we lectured in Trieste in 2003 and Luminy in 2004. The courses were based on the books [1, 3,

### and the diagram commutes, which means that every fibre E p is mapped to {p} V.

Part III: Differential geometry (Michaelmas 2004) Alexei Kovalev (A.G.Kovalev@dpmms.cam.ac.uk) 2 Vector bundles. Definition. Let B be a smooth manifold. A manifold E together with a smooth submersion 1

### 1 Hermitian symmetric spaces: examples and basic properties

Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................

### 30 Surfaces and nondegenerate symmetric bilinear forms

80 CHAPTER 3. COHOMOLOGY AND DUALITY This calculation is useful! Corollary 29.4. Let p, q > 0. Any map S p+q S p S q induces the zero map in H p+q ( ). Proof. Let f : S p+q S p S q be such a map. It induces

### MASLOV INDEX. Contents. Drawing from [1]. 1. Outline

MASLOV INDEX J-HOLOMORPHIC CURVE SEMINAR MARCH 2, 2015 MORGAN WEILER 1. Outline 1 Motivation 1 Definition of the Maslov Index 2 Questions Demanding Answers 2 2. The Path of an Orbit 3 Obtaining A 3 Independence

### Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture)

Building Geometric Structures: Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover

### Topic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016

Topic: First Chern classes of Kähler manifolds itchell Faulk Last updated: April 23, 2016 We study the first Chern class of various Kähler manifolds. We only consider two sources of examples: Riemann surfaces

### Topology of Lie Groups

Topology of Lie Groups Lecture 1 In this seminar talks, to begin with, we plan to present some of the classical results on the topology of Lie groups and (homogeneous spaces). The basic assumption is that

### k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim

### 7.3 Singular Homology Groups

184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the

### IGA Lecture I: Introduction to G-valued moment maps

IGA Lecture I: Introduction to G-valued moment maps Adelaide, September 5, 2011 Review: Hamiltonian G-spaces Let G a Lie group, g = Lie(G), g with co-adjoint G-action denoted Ad. Definition A Hamiltonian

### arxiv:math-ph/ v1 31 May 2000

An Elementary Introduction to Groups and Representations arxiv:math-ph/0005032v1 31 May 2000 Author address: Brian C. Hall University of Notre Dame, Department of Mathematics, Notre Dame IN 46556 USA E-mail

### Flat connections on 2-manifolds Introduction Outline:

Flat connections on 2-manifolds Introduction Outline: 1. (a) The Jacobian (the simplest prototype for the class of objects treated throughout the paper) corresponding to the group U(1)). (b) SU(n) character

### Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

### Topological K-theory, Lecture 3

Topological K-theory, Lecture 3 Matan Prasma March 2, 2015 1 Applications of the classification theorem continued Let us see how the classification theorem can further be used. Example 1. The bundle γ

### A Crash Course of Floer Homology for Lagrangian Intersections

A Crash Course of Floer Homology for Lagrangian Intersections Manabu AKAHO Department of Mathematics Tokyo Metropolitan University akaho@math.metro-u.ac.jp 1 Introduction There are several kinds of Floer

### Delzant s Garden. A one-hour tour to symplectic toric geometry

Delzant s Garden A one-hour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem

### THE H-PRINCIPLE, LECTURE 14: HAEFLIGER S THEOREM CLASSIFYING FOLIATIONS ON OPEN MANIFOLDS

THE H-PRINCIPLE, LECTURE 14: HAELIGER S THEOREM CLASSIYING OLIATIONS ON OPEN MANIOLDS J. RANCIS, NOTES BY M. HOYOIS In this lecture we prove the following theorem: Theorem 0.1 (Haefliger). If M is an open

### Elementary linear algebra

Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The

### LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups

LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are