1 Piecewise linear manifolds 7

Contents

Introduction 1

1 Piecewise linear manifolds 7

1.1 Simplicial complexes . . . . 7

1.2 Polyhedra . . . 10

1.3 Piecewise linear manifolds . . . 13

1.4 Triangulations of smooth manifolds . . . 15

1.5 Dierential forms on PL manifolds . . . 17

2 Piecewise linear symplectic manifolds 19 2.1 Piecewise constant symplectic structures . . . 20

2.2 Darboux's property . . . 21

2.2.1 In dimension two . . . 22

2.2.2 In other dimensions . . . 23

2.3 Symplectic triangulations . . . 26

3 Triangulation of volume forms 29 3.1 Why not Moser ? . . . 30

3.2 Relative Moser . . . 31

3.3 Agreement near the skeleton . . . 32

3.3.1 Near vertices . . . 32

3.3.2 Near the entire skeleton . . . 34

3.4 Several Consequences . . . 36

3.5 Interpolation . . . 38

3.6 Alternative proof in dimension two . . . 41

3.6.1 Odd triangulations . . . 42

3.6.2 Agreement at vertices . . . 43

3.6.3 Relative Moser . . . 44

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4 Symplectic jiggling lemma 47

4.1 Crystalline subdivision of a complex . . . 48

4.2 Symplectic jiggling lemma . . . 50

4.2.1 Statement of the lemma . . . 50

4.2.2 Proof of the symplectic jiggling lemma . . . 53

4.3 Construction of PL symplectic manifolds . . . 57

5 Symplectic submanifold of a triangulated manifold 61 5.1 Transverse triangulations . . . 62

5.2 PL approximation of a symplectic submanifold . . . 66

5.3 Characteristic foliation . . . 69

5.4 Tubular neighborhood of a symplectic submanifold . . . 70

5.4.1 Symplectic structure on the total space of a vector bundle . . . 70

5.4.2 Symplectic universal bundle . . . 71

5.4.3 Symplectic normal bundle . . . 74

6 Perspectives 75