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Singular Foliations and Their Holonomy Singular foliations and their holonomy Iakovos Androulidakis Department of Mathematics, University of Athens Joint work with M. Zambon (Univ. Aut´onomaMadrid - ICMAT) and G. Skandalis (Paris 7 Denis Diderot) Z¨urich,December 2012 I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 1 / 24 Summary 1 Introduction Regular Foliations Holonomy groupoid Reeb stability 2 Singular foliations What is a singular foliation? Holonomy groupoid Essential isotropy Holonomy map 3 Linearization I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 2 / 24 In other words: There is an open cover of M by foliation charts of the p q form Ω = U × T, where U ⊆ R and T ⊆ R . T is the transverse direction and U is the longitudinal or leafwise direction. The change of charts is of the form f(u, t) = (g(u, t), h(t)). Viewpoint 2: Frobenius theorem Consider the unique C (M)-module F of vector fields tangent to leaves. 1 Fact: F = Cc (M, F) and [F, F] ⊆ F. 1 Introduction Regular Foliations 1.1 Definition: Foliation (regular) Viewpoint 1: Partition to connected submanifolds. Local picture: I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 3 / 24 The change of charts is of the form f(u, t) = (g(u, t), h(t)). Viewpoint 2: Frobenius theorem Consider the unique C (M)-module F of vector fields tangent to leaves. 1 Fact: F = Cc (M, F) and [F, F] ⊆ F. 1 Introduction Regular Foliations 1.1 Definition: Foliation (regular) Viewpoint 1: Partition to connected submanifolds. Local picture: In other words: There is an open cover of M by foliation charts of the p q form Ω = U × T, where U ⊆ R and T ⊆ R . T is the transverse direction and U is the longitudinal or leafwise direction. I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 3 / 24 Viewpoint 2: Frobenius theorem Consider the unique C (M)-module F of vector fields tangent to leaves. 1 Fact: F = Cc (M, F) and [F, F] ⊆ F. 1 Introduction Regular Foliations 1.1 Definition: Foliation (regular) Viewpoint 1: Partition to connected submanifolds. Local picture: In other words: There is an open cover of M by foliation charts of the p q form Ω = U × T, where U ⊆ R and T ⊆ R . T is the transverse direction and U is the longitudinal or leafwise direction. The change of charts is of the form f(u, t) = (g(u, t), h(t)). I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 3 / 24 Introduction Regular Foliations 1.1 Definition: Foliation (regular) Viewpoint 1: Partition to connected submanifolds. Local picture: In other words: There is an open cover of M by foliation charts of the p q form Ω = U × T, where U ⊆ R and T ⊆ R . T is the transverse direction and U is the longitudinal or leafwise direction. The change of charts is of the form f(u, t) = (g(u, t), h(t)). Viewpoint 2: Frobenius theorem Consider the unique C (M)-module F of vector fields tangent to leaves. 1 Fact: F = Cc (M, F) and [F, F] ⊆ F. I. Androulidakis (Athens) Singular1 foliations and their holonomy Z¨urich,December 2012 3 / 24 p + q degrees of freedom for x; then p degrees of freedom for y. A neighborhood of (x, x0) where x 2 W = U × T and x0 2 W 0 = U0 × T 0 should be of the form U × U0 × T: we need an identification of T with T 0. (Here T, T 0 are local transversals.) Definition A holonomy of (M, F) is a diffeomorphism h : T ! T 0 such that t, h(t) are in the same leaf for all t 2 T. Introduction Holonomy groupoid Holonomy groupoid of a regular foliation Holonomy We wish to put a smooth structure on the equivalence relation 2 f(x, y) 2 M : Lx = Lyg What is the dimension of this manifold? I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 4 / 24 A neighborhood of (x, x0) where x 2 W = U × T and x0 2 W 0 = U0 × T 0 should be of the form U × U0 × T: we need an identification of T with T 0. (Here T, T 0 are local transversals.) Definition A holonomy of (M, F) is a diffeomorphism h : T ! T 0 such that t, h(t) are in the same leaf for all t 2 T. Introduction Holonomy groupoid Holonomy groupoid of a regular foliation Holonomy We wish to put a smooth structure on the equivalence relation 2 f(x, y) 2 M : Lx = Lyg What is the dimension of this manifold? p + q degrees of freedom for x; then p degrees of freedom for y. I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 4 / 24 Definition A holonomy of (M, F) is a diffeomorphism h : T ! T 0 such that t, h(t) are in the same leaf for all t 2 T. Introduction Holonomy groupoid Holonomy groupoid of a regular foliation Holonomy We wish to put a smooth structure on the equivalence relation 2 f(x, y) 2 M : Lx = Lyg What is the dimension of this manifold? p + q degrees of freedom for x; then p degrees of freedom for y. A neighborhood of (x, x0) where x 2 W = U × T and x0 2 W 0 = U0 × T 0 should be of the form U × U0 × T: we need an identification of T with T 0. (Here T, T 0 are local transversals.) I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 4 / 24 Introduction Holonomy groupoid Holonomy groupoid of a regular foliation Holonomy We wish to put a smooth structure on the equivalence relation 2 f(x, y) 2 M : Lx = Lyg What is the dimension of this manifold? p + q degrees of freedom for x; then p degrees of freedom for y. A neighborhood of (x, x0) where x 2 W = U × T and x0 2 W 0 = U0 × T 0 should be of the form U × U0 × T: we need an identification of T with T 0. (Here T, T 0 are local transversals.) Definition A holonomy of (M, F) is a diffeomorphism h : T ! T 0 such that t, h(t) are in the same leaf for all t 2 T. I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 4 / 24 If W = U × T and W 0 = U0 × T 0, in their intersection u0 = g(u, t) and 0 t = h(t) by definition of the chart changes. The map h = hW 0,W is a holonomy. Let γ :[0, 1] ! M be a smooth path in a leaf. Cover γ by foliation charts Wi = Ui × Ti(1 6 i 6 n). Consider the composition h(γ) = hWn,Wn-1 ◦ ... ◦ hW2,W1 Definition The holonomy of the path γ is the germ of h(γ). Fact: Path holonomy depends only on the homotopy class of the path! Introduction Holonomy groupoid Examples of holonomies Take W = U × T. Then idT is a holonomy. If h is a holonomy, h-1 is a holonomy. The composition of holonomies is a holonomy. (Holonomies form a pseudogroup.) I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 5 / 24 Let γ :[0, 1] ! M be a smooth path in a leaf. Cover γ by foliation charts Wi = Ui × Ti(1 6 i 6 n). Consider the composition h(γ) = hWn,Wn-1 ◦ ... ◦ hW2,W1 Definition The holonomy of the path γ is the germ of h(γ). Fact: Path holonomy depends only on the homotopy class of the path! Introduction Holonomy groupoid Examples of holonomies Take W = U × T. Then idT is a holonomy. If h is a holonomy, h-1 is a holonomy. The composition of holonomies is a holonomy. (Holonomies form a pseudogroup.) If W = U × T and W 0 = U0 × T 0, in their intersection u0 = g(u, t) and 0 t = h(t) by definition of the chart changes. The map h = hW 0,W is a holonomy. I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 5 / 24 Introduction Holonomy groupoid Examples of holonomies Take W = U × T. Then idT is a holonomy. If h is a holonomy, h-1 is a holonomy. The composition of holonomies is a holonomy. (Holonomies form a pseudogroup.) If W = U × T and W 0 = U0 × T 0, in their intersection u0 = g(u, t) and 0 t = h(t) by definition of the chart changes. The map h = hW 0,W is a holonomy. Let γ :[0, 1] ! M be a smooth path in a leaf. Cover γ by foliation charts Wi = Ui × Ti(1 6 i 6 n). Consider the composition h(γ) = hWn,Wn-1 ◦ ... ◦ hW2,W1 Definition The holonomy of the path γ is the germ of h(γ). Fact: Path holonomy depends only on the homotopy class of the path! I. Androulidakis (Athens) Singular foliations and their holonomy Z¨urich,December 2012 5 / 24 H(F) is a Lie groupoid. Its Lie algebroid is F. Its orbits are the leaves. H(F) is the smallest possible smooth groupoid over F. Introduction Holonomy groupoid The holonomy groupoid Definition The holonomy groupoid is H(F) = f(x, y, h(γ))g where γ is a path in a leaf joining x to y. Manifold structure. If W = U × T and W 0 = U0 × T 0 are charts and h : T ! T 0 path-holonomy, get chart 0 Ωh = U × U × T Groupoid structure.
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