An introduction to orbifolds

Transcription

1 An introduction to orbifolds Joan Porti UAB Subdivide and Tile: Triangulating spaces for understanding the world Lorentz Center November 009 An introduction to orbifolds p.1/0

2 Motivation Γ < Isom(R n ) or H n discrete and acts properly discontinuously (e.g. a group of symmetries of a tessellation). If Γ has no fixed points Γ\R n is a manifold. If Γ has fixed points Γ\R n is an orbifold. An introduction to orbifolds p./0

3 Motivation Γ < Isom(R n ) or H n discrete and acts properly discontinuously (e.g. a group of symmetries of a tessellation). If Γ has no fixed points Γ\R n is a manifold. If Γ has fixed points Γ\R n is an orbifold. (there are other notions of orbifold in algebraic geometry, string theory or using grupoids) An introduction to orbifolds p./0

4 Examples: tessellations of Euclidean plane Γ = (x, y) (x + 1, y), (x, y) (x, y + 1) = Z Γ\R = T = S 1 S 1 An introduction to orbifolds p.3/0

5 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order ) An introduction to orbifolds p.3/0

6 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order ) An introduction to orbifolds p.3/0

7 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order ) An introduction to orbifolds p.3/0

8 Example: tessellations of hyperbolic plane Rotations of angle π, π/ and π/3 around vertices (order, 4, and 6) An introduction to orbifolds p.4/0

9 Example: tessellations of hyperbolic plane Rotations of angle π, π/ and π/3 around vertices (order, 4, and 6) 4 6 An introduction to orbifolds p.4/0

10 Definition Informal Definition An orbifold O is a metrizable topological space equipped with an atlas modelled on R n /Γ, Γ < O(n) finite, with some compatibility condition. We keep track of the local action of Γ < O(n). An introduction to orbifolds p.5/0

11 Definition Informal Definition An orbifold O is a metrizable topological space equipped with an atlas modelled on R n /Γ, Γ < O(n) finite, with some compatibility condition. We keep track of the local action of Γ < O(n). Singular (or branching) locus: Σ= points modelled on F ix(γ)/γ. Γ x (the minimal Γ): isotropy group of a point x O. O underlying topological space (possibly not a manifold). An introduction to orbifolds p.5/0

12 Definition Formal Definition An orbifold O is a metrizable top. space with a (maximal) atlas Ui = O, Γ i < O(n) Ũ i R n is Γ i -invariant φ i : U i = Ũ i /Γ i homeo Ũ i R n {U i, Ũi, Γ i, φ i } U i O = U i /Γ i An introduction to orbifolds p.5/0

13 Definition Formal Definition An orbifold O is a metrizable top. space with a (maximal) atlas Ui = O, Γ i < O(n) Ũ i R n is Γ i -invariant φ i : U i = Ũ i /Γ i homeo {U i, Ũi, Γ i, φ i } If y U i U j, then there is U k s.t. y U k U i U j and Ũ i R n i : Γ k Γ i i : Ũ k Ũ i, diffeo with the image, U i O = U i /Γ i i(γ x) = i (γ) i(x) Γ x = x U i Γ i isotropy group of a point Σ O ={ x Γ x 1} An introduction to orbifolds p.5/0

14 Dimension Finite subgroups of O(): cyclic group of rotations with n elements reflexion along a line dihedral group rots. and reflex. Local picture of -orbifolds: n n Γ x cyclic order n Γ x has elements Γ x dihedral An introduction to orbifolds p.6/0

15 Dimension Finite subgroups of O(): cyclic group of rotations with n elements reflexion along a line dihedral group rots. and reflex. General picture in dim m 4 n 3 n 1 m 1 n m 3 O surface, Σ O = O and points m An introduction to orbifolds p.6/0

16 Dimension 3 (loc.orientable) Finite subgroups of SO(3) (all elements are rotations): C n D n T 1 O 4 I 60 cyclic dihedral tetrahedral octahedral icosahedral O dim 3 and orientable: O = manifold, Σ O trivalent graph C n D n T 1 O 4 I 60 Σ O locally: n n An introduction to orbifolds p.7/0

17 Dimension 3 (loc.orientable) O dim 3 and orientable: O = manifold, Σ O trivalent graph C n D n T 1 O 4 I 60 Σ O locally: n n Non orientable case: combine this with reflections along planes and antipodal map: a(x, y, z) = ( x, y, z) R 3 /a = cone on RP, is not a manifold. In dim 4 and larger, O orientable and O possibly not a manifold. An introduction to orbifolds p.7/0

18 More examples: tessellation by cubes Z 3 translation group, R 3 /Z 3 = S 1 S 1 S 1 But we can also consider other groups An introduction to orbifolds p.8/0

19 More examples: tessellation by cubes O = R 3 / reflections on the sides of the cube O is the cube and Σ O boundary of the cube x in a face Γ x = Z/Z reflexion x in an edge Γ x = (Z/Z) x in a vertex Γ x = (Z/Z) 3 An introduction to orbifolds p.8/0

20 More examples: tessellation by cubes Consider the group generated by order rotations around axis as in: An introduction to orbifolds p.8/0

21 More examples: tessellation by cubes Consider the group generated by order rotations around axis as in: O = S 3 Σ O = Borromean rings. Γ x = Z/Z acting by rotations. An introduction to orbifolds p.8/0

22 More examples: tessellation by cubes R 3 /{ Full isometry group of the tessellation } x in a face Γ x = Z/Z reflexion x in an edge Γ x = dihedral (extension by reflections of cyclic group of rotations) x in a vertex Γ x = extension by reflections of dihedral, T 1 or O 4 An introduction to orbifolds p.8/0

23 More examples: hyperbolic tessellation An introduction to orbifolds p.9/0

24 More examples: hyperbolic tessellation An introduction to orbifolds p.9/0

25 Fundamental domain If Γ acts on R or H n Dirichlet domain: Voronoi cell of the orbit of a point x 0 with Γ x0 = 1. γ 1 x 0 γ 1 3 x 0 γ 1 1 x 0 x 0 γ 1 x 0 γ 3 x 0 γ x 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds p.10/0

26 Fundamental domain If Γ acts on R or H n Dirichlet domain: Voronoi cell of the orbit of a point x 0 with Γ x0 = 1. x 0 γ 1 x 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds p.10/0

27 Fundamental domain If Γ acts on R or H n Dirichlet domain: Voronoi cell of the orbit of a point x 0 with Γ x0 = 1. x 0 γ 1 x 0 γ x 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds p.10/0

28 Fundamental domain If Γ acts on R or H n Dirichlet domain: Voronoi cell of the orbit of a point x 0 with Γ x0 = 1. x 0 γ 1 x 0 γ 3 x 0 γ x 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds p.10/0

29 Fundamental domain If Γ acts on R or H n Dirichlet domain: Voronoi cell of the orbit of a point x 0 with Γ x0 = 1. γ 1 1 x 0 x 0 γ 1 x 0 γ 3 x 0 γ x 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds p.10/0

30 Fundamental domain If Γ acts on R or H n Dirichlet domain: Voronoi cell of the orbit of a point x 0 with Γ x0 = 1. γ 1 x 0 γ 1 1 x 0 x 0 γ 1 x 0 γ 3 x 0 γ x 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds p.10/0

31 Fundamental domain If Γ acts on R or H n Dirichlet domain: Voronoi cell of the orbit of a point x 0 with Γ x0 = 1. γ 1 x 0 γ 1 3 x 0 γ 1 1 x 0 x 0 γ 1 x 0 γ 3 x 0 γ x 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds p.10/0

32 Fundamental domain If Γ acts on R or H n Dirichlet domain: Voronoi cell of the orbit of a point x 0 with Γ x0 = 1. γ 1 x 0 γ 1 3 x 0 γ 1 1 x 0 x 0 γ 1 x 0 γ 3 x 0 γ x 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds p.10/0

33 Structures on orbifolds Can define geometric structures on orbifolds by taking equivariant definitions on charts Ũi, Γ i. A Riemanian metric on the charts (Ũ i, Γ i ) requires: Ũ i has a Riemannian metric coordinate changes are isometries Γ i acts isometrically on Ũi e.g. Analytic, flat, hyperbolic, etc... An introduction to orbifolds p.11/0

34 Structures on orbifolds Can define geometric structures on orbifolds by taking equivariant definitions on charts Ũi, Γ i. A Riemanian metric on the charts (Ũ i, Γ i ) requires: Ũ i has a Riemannian metric coordinate changes are isometries Γ i acts isometrically on Ũi e.g. Analytic, flat, hyperbolic, etc... 4 has a flat metric (Euclidean) has a metric of curv 1 (hyperbolic) 6 An introduction to orbifolds p.11/0

35 Structures on orbifolds Can define geometric structures on orbifolds by taking equivariant definitions on charts Ũi, Γ i has a flat metric (Euclidean) has a metric of curv 1 (hyperbolic) An introduction to orbifolds p.11/0

36 Coverings p : O 0 O 1 is an orbifold covering if Every x O 1 is in some U O 1 s.t. if V = component of p 1 (U): then Ṽ V p U is a chart for U V = Ṽ /Γ 0 Ṽ R n Γ 0 < Γ 1 p U = Ũ/Γ 1 Γ 0 equiv Ũ R n Branched coverings can be seen as orbifold coverings If Γ acts properly discontinuously on M manifold then M M/Γ is an orbifold covering. An introduction to orbifolds p.1/0

37 Coverings Branched coverings can be seen as orbifold coverings If Γ acts properly discontinuously on M manifold then M M/Γ is an orbifold covering. Examples An introduction to orbifolds p.1/0

38 Coverings Branched coverings can be seen as orbifold coverings If Γ acts properly discontinuously on M manifold then M M/Γ is an orbifold covering. Examples An introduction to orbifolds p.1/0

39 Good and bad Definition O is good if O = M/Γ Γ acts properly discontinuously on a manifold M O is good iff O has a covering that is a manifold. Question When is an orbifold good? An introduction to orbifolds p.13/0

40 Good and bad Definition O is good if O = M/Γ Γ acts properly discontinuously on a manifold M O is good iff O has a covering that is a manifold. Question n When is an orbifold good? p are BAD q Teardrop T(n) Spindle S(p, q) (p q) An introduction to orbifolds p.13/0

41 Good and bad Definition O is good if O = M/Γ Γ acts properly discontinuously on a manifold M O is good iff O has a covering that is a manifold. n p are BAD Teardrop T(n) Spindle S(p, q) (p q) Those and their nonorientable quotients are the only bad -orbifolds q An introduction to orbifolds p.13/0

42 Fundamental group Def: A loop based at x O \ Σ O : γ : [0, 1] O such that γ(0) = γ(1) = x with a choice of lifts at branchings: Ũ γ 3 : 1 γ Define homotopies as continuous 1-parameter families of paths. π 1 (O, x) = { loops based at x up to homotopy relative to x} U An introduction to orbifolds p.14/0

43 Fundamental group π 1 (O, x) = { loops based at x up to homotopy relative to x} π 1 (D /rotation order n) = Z/nZ Ũ γ 3 : 1 U γ γ 3 = 1 An introduction to orbifolds p.14/0

44 Fundamental group π 1 (O, x) = { loops based at x up to homotopy relative to x} Seifert-Van Kampen theorem (Haëfliger): If O = U V, U V conected, then: π 1 (O) = π 1 (U) π1 (U V ) π 1 (V ) π 1 O = free product of π 1 (U) and π 1 (V ) quotiented by π 1 (U V ) An introduction to orbifolds p.14/0

45 Fundamental group π 1 (O, x) = { loops based at x up to homotopy relative to x} Seifert-Van Kampen theorem (Haëfliger): If O = U V, U V conected, then: π 1 (O) = π 1 (U) π1 (U V ) π 1 (V ) π 1 O = free product of π 1 (U) and π 1 (V ) quotiented by π 1 (U V ) π 1 (T(n)) = {1} n n U V U V An introduction to orbifolds p.14/0

46 Fundamental group π 1 (O, x) = { loops based at x up to homotopy relative to x} Seifert-Van Kampen theorem (Haëfliger): If O = U V, U V conected, then: π 1 (O) = π 1 (U) π1 (U V ) π 1 (V ) π 1 O = free product of π 1 (U) and π 1 (V ) quotiented by π 1 (U V ) π 1 (S(p, q)) = Z/gcd(p, q)z p p q q U V U V An introduction to orbifolds p.14/0

47 Universal covering Universal covering: Õ O such that every other covering O O: Õ O O Existence: Õ = {rel. homotopy classes of paths starting at x} An introduction to orbifolds p.15/0

48 Universal covering Universal covering: Õ O such that every other covering O O: Õ O O Existence: Õ = {rel. homotopy classes of paths starting at x} π 1 (O) = deck transformation group of Õ O. π 1 (T(n)) = {1} and π 1 (S(p, q)) = Z/ gcd(p, q)z, hence T(n) and S(p, q), p q, are bad. An introduction to orbifolds p.15/0

49 Developable orbifolds Theorem 1. If an orbifold has a metric of constant curvature, then it is good.. If an orbifold has a metric of nonpositive curvature, then it is good. Proof 1: use developing maps Ũ H n, S n, R n Proof : use developing maps, convexity, uniqueness of geodesics. An introduction to orbifolds p.16/0

50 Developable orbifolds Theorem 1. If an orbifold has a metric of constant curvature, then it is good.. If an orbifold has a metric of nonpositive curvature, then it is good. Proof 1: use developing maps Ũ H n, S n, R n Proof : use developing maps, convexity, uniqueness of geodesics. Corollary: All orientable -orbifolds other than T(n) and S(p, q), p q have a constant curvature metric, hence are good. Can put an orbifold metric of constant curvature by using polygons. An introduction to orbifolds p.16/0

51 dim example: turnovers The triangle β α γ is hyperbolic if α + β + γ < π Euclidean if α + β + γ = π spherical if α + β + γ > π An introduction to orbifolds p.17/0

52 dim example: turnovers The triangle β α γ is hyperbolic if α + β + γ < π Euclidean if α + β + γ = π spherical if α + β + γ > π Glue two triangles along the boundaries, set α = π n 1, β = π n, γ = π n 3, O = S, Σ O =three points, cyclic isotropy groups of orders n 1, n, n 3. The orbifold n n 1 n 3 is hyperbolic if 1 n n + 1 n 3 < 1 Euclidean if 1 n n + 1 n 3 = 1 spherical if 1 n n + 1 n 3 > 1 An introduction to orbifolds p.17/0

53 dim example: turnovers The triangle β α γ is hyperbolic if α + β + γ < π Euclidean if α + β + γ = π spherical if α + β + γ > π Glue two triangles along the boundaries, set α = π n 1, β = π n, γ = π n 3, O = S, Σ O =three points, cyclic isotropy groups of orders n 1, n, n 3. The orbifold n n 1 n 3 is The metric on S is singular, but not in O hyperbolic if 1 n n + 1 n 3 < 1 Euclidean if 1 n n + 1 n 3 = 1 spherical if 1 n n + 1 n 3 > 1 An introduction to orbifolds p.17/0

54 Euler characteristic χ(o) = e ( 1) dim e 1 Γ e The sum runs over the cells of a cellulation of O that preserves the stratification of the branching locus. An introduction to orbifolds p.18/0

55 Euler characteristic χ(o) = e ( 1) dim e 1 Γ e The sum runs over the cells of a cellulation of O that preserves the stratification of the branching locus. Properties: If O O is a covering of degree n χ(o) = nχ(o ) Gauss-Bonnet formula. If dim O =, then: K = πχ(o) where K = curvature. O An introduction to orbifolds p.18/0

56 Ricci flow on two orbifolds Normalized Ricci flow: g t = Ric+ n r g g Riemmannian metric, r = O scal/ 1 average scalar curvature O Ric Ricci curvature. In dim, Ric = K g. So write g t = e u g 0 for some function u = u(x, t). The conformal class is preserved and t u = e u g0 u An introduction to orbifolds p.19/0

57 Ricci flow on two orbifolds Normalized Ricci flow: g t = Ric+ n r g Hamilton, Chow, Wu, Chen-Lu-Tian: Either it converges to a metric of constant curvature, or to a gradient soliton on T(n) or S(p, q) n p Gradient soliton: g t = a t φ tg 0, with t φ t = grad(f) q An introduction to orbifolds p.19/0

58 Dimension 3 A 3-orbifold is good iff it does not contain bad -suborbifolds. An introduction to orbifolds p.0/0

59 Dimension 3 A 3-orbifold is good iff it does not contain bad -suborbifolds. Thurston s orbifold theorem If O has no bad -suborbifolds, then O decomposes canonically into locally homogeneous pieces O Locally homogeneous, if O = M/Γ, M = homogeneous manifold eg. R 3, H 3, S 3, H R, PSL (R)... Canonical decomposition: 1. Orbifold connected sum: (O 1 \ B 3 /Γ) S /Γ (O \ B 3 /Γ). Cut along T /Γ π 1 -injective in O (orbifold JSJ-theory) An introduction to orbifolds p.0/0

60 Dimension 3 A 3-orbifold is good iff it does not contain bad -suborbifolds. Thurston s orbifold theorem If O has no bad -suborbifolds, then O decomposes canonically into locally homogeneous pieces O Locally homogeneous, if O = M/Γ, M = homogeneous manifold eg. R 3, H 3, S 3, H R, PSL (R)... Canonical decomposition: 1. Orbifold connected sum: (O 1 \ B 3 /Γ) S /Γ (O \ B 3 /Γ). Cut along T /Γ π 1 -injective in O (orbifold JSJ-theory) Õ = diff R 3, S 3 or infinite connected sums. An introduction to orbifolds p.0/0

61 Dimension 3 A 3-orbifold is good iff it does not contain bad -suborbifolds. O = O 1 #O, O 1, O hyperbolic, then Õ infinite connected sum: An introduction to orbifolds p.0/0

62 Dimension 3 A 3-orbifold is good iff it does not contain bad -suborbifolds. O = O 1 #O, O 1, O hyperbolic, then Õ infinite connected sum: An introduction to orbifolds p.0/0

63 Dimension 3 A 3-orbifold is good iff it does not contain bad -suborbifolds. O = O 1 #O, O 1, O hyperbolic, then Õ infinite connected sum: An introduction to orbifolds p.0/0

64 Dimension 3 A 3-orbifold is good iff it does not contain bad -suborbifolds. O = O 1 #O, O 1, O hyperbolic, then Õ infinite connected sum: An introduction to orbifolds p.0/0