An introduction to orbifolds Joan Porti UAB Subdivide and Tile: Triangulating spaces for understanding the world Lorentz Center November 2009 An introduction to orbifolds – p.1/20 Motivation • Γ < Isom(Rn) or Hn discrete and acts properly discontinuously (e.g. a group of symmetries of a tessellation). • If Γ has no fixed points ⇒ Γ\Rn is a manifold. • If Γ has fixed points ⇒ Γ\Rn is an orbifold. An introduction to orbifolds – p.2/20 Motivation • Γ < Isom(Rn) or Hn discrete and acts properly discontinuously (e.g. a group of symmetries of a tessellation). • If Γ has no fixed points ⇒ Γ\Rn is a manifold. • If Γ has fixed points ⇒ Γ\Rn is an orbifold. ··· (there are other notions of orbifold in algebraic geometry, string theory or using grupoids) An introduction to orbifolds – p.2/20 Examples: tessellations of Euclidean plane Γ= h(x, y) → (x + 1, y), (x, y) → (x, y + 1)i =∼ Z2 Γ\R2 =∼ T 2 = S1 × S1 An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) 2 2 An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) 2 2 2 2 2 2 2 2 2 2 2 2 An introduction to orbifolds – p.3/20 Example: tessellations of hyperbolic plane Rotations of angle π, π/2 and π/3 around vertices (order 2, 4, and 6) An introduction to orbifolds – p.4/20 Example: tessellations of hyperbolic plane Rotations of angle π, π/2 and π/3 around vertices (order 2, 4, and 6) 2 4 2 6 An introduction to orbifolds – p.4/20 Definition Informal Definition • An orbifold O is a metrizable topological space equipped with an atlas modelled on Rn/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.5/20 Definition Informal Definition • An orbifold O is a metrizable topological space equipped with an atlas modelled on Rn/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. • Γx (the minimal Γ): isotropy group of a point x ∈ O. • |O| underlying topological space (possibly not a manifold). An introduction to orbifolds – p.5/20 Definition Formal Definition • An orbifold O is a metrizable top. space with a (maximal) atlas {Ui, Ui, Γi,φi} Ui = O, Γi < O(n) e S n Ui ⊂ R is Γi-invariant φei : Ui =∼ Ui/Γi homeo e n Ui ⊂ R e Ui ⊂ O =∼ Ui/Γi An introduction to orbifolds – p.5/20 Definition Formal Definition • An orbifold O is a metrizable top. space with a (maximal) atlas {Ui, Ui, Γi,φi} Ui = O, Γi < O(n) e S n Ui ⊂ R is Γi-invariant If y ∈ Ui ∩ Uj, then there is Uk φei : Ui =∼ Ui/Γi homeo s.t. y ∈ Uk ⊂ Ui ∩ Uj and e n Ui ⊂ R • i∗ :Γk ֒→ Γi ,e • i : Uk ֒→ Ui, diffeo with the image • i(γe· x)=ei∗(γ) · i(x) Ui ⊂ O =∼ Ui/Γi Γx = Γi isotropy group of a point ΣO ={ x | Γx 6= 1} x∈TUi An introduction to orbifolds – p.5/20 Dimension 2 Finite subgroups of O(2): cyclic group of rotations reflexion along a line dihedral group with n elements rots. and reflex. Local picture of 2-orbifolds: n n Γx cyclic order n Γx has 2 elements Γx dihedral An introduction to orbifolds – p.6/20 Dimension 2 Finite subgroups of O(2): cyclic group of rotations reflexion along a line dihedral group with n elements rots. and reflex. General picture in dim 2 2m4 n3 2m3 n1 2m1 n2 • |O2| surface, 2m2 2 ΣO2 = ∂|O | and points An introduction to orbifolds – p.6/20 Dimension 3 (loc.orientable) Finite subgroups of SO(3) (all elements are rotations): Cn D2n T12 O24 I60 cyclic dihedral tetrahedral octahedral icosahedral • O dim 3 and orientable: |O| = manifold, ΣO trivalent graph Cn D2n T12 O24 I60 n 3 4 5 ΣO locally: n 2 2 2 3 2 3 2 3 An introduction to orbifolds – p.7/20 Dimension 3 (loc.orientable) • O dim 3 and orientable: |O| = manifold, ΣO trivalent graph Cn D2n T12 O24 I60 n 3 4 5 ΣO locally: n 2 2 2 3 2 3 2 3 • Non orientable case: combine this with reflections along planes and antipodal map: a(x, y, z)=(−x, −y, −z) • R3/a = cone on RP2, is not a manifold. • In dim 4 and larger, ∃O orientable and O possibly not a manifold. An introduction to orbifolds – p.7/20 More examples: tessellation by cubes • Z3 translation group, R3/Z3 = S1 × S1 × S1 • But we can also consider other groups An introduction to orbifolds – p.8/20 More examples: tessellation by cubes • O = R3/hreflections on the sides of the cubei |O| is the cube and ΣO boundary of the cube • x in a face ⇒ Γx = Z/2Z reflexion 2 • x in an edge⇒ Γx =(Z/2Z) 3 • x in a vertex ⇒ Γx =(Z/2Z) An introduction to orbifolds – p.8/20 More examples: tessellation by cubes Consider the group generated by order 2 rotations around axis as in: An introduction to orbifolds – p.8/20 More examples: tessellation by cubes Consider the group generated by order 2 rotations around axis as in: 2 2 2 3 • |O| = S ΣO = Borromean rings. Γx =∼ Z/2Z acting by rotations. An introduction to orbifolds – p.8/20 More examples: tessellation by cubes R3/{ Full isometry group of the tessellation } • x in a face ⇒ Γx = Z/2Z reflexion • x in an edge ⇒ Γx = dihedral (extension by reflections of cyclic group of rotations) • x in a vertex ⇒ Γx = extension by reflections of dihedral, T12 or O24 An introduction to orbifolds – p.8/20 More examples: hyperbolic tessellation An introduction to orbifolds – p.9/20 More examples: hyperbolic tessellation 4 4 4 An introduction to orbifolds – p.9/20 Fundamental domain • If Γ acts on R or Hn Dirichlet domain: Voronoi cell of the orbit of a point x0 with Γx0 = 1. −1 γ2 x0 −1 γ3 x0 −1 γ1 x0 x0 γ1x0 γ x γ3x0 2 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds – p.10/20 Fundamental domain • If Γ acts on R or Hn Dirichlet domain: Voronoi cell of the orbit of a point x0 with Γx0 = 1. x0 γ1x0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds – p.10/20 Fundamental domain • If Γ acts on R or Hn Dirichlet domain: Voronoi cell of the orbit of a point x0 with Γx0 = 1. x0 γ1x0 γ2x0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds – p.10/20 Fundamental domain • If Γ acts on R or Hn Dirichlet domain: Voronoi cell of the orbit of a point x0 with Γx0 = 1. x0 γ1x0 γ x γ3x0 2 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds – p.10/20 Fundamental domain • If Γ acts on R or Hn Dirichlet domain: Voronoi cell of the orbit of a point x0 with Γx0 = 1. −1 γ1 x0 x0 γ1x0 γ x γ3x0 2 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds – p.10/20 Fundamental domain • If Γ acts on R or Hn Dirichlet domain: Voronoi cell of the orbit of a point x0 with Γx0 = 1. −1 γ2 x0 −1 γ1 x0 x0 γ1x0 γ x γ3x0 2 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds – p.10/20 Fundamental domain • If Γ acts on R or Hn Dirichlet domain: Voronoi cell of the orbit of a point x0 with Γx0 = 1. −1 γ2 x0 −1 γ3 x0 −1 γ1 x0 x0 γ1x0 γ x γ3x0 2 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds – p.10/20 Fundamental domain • If Γ acts on R or Hn Dirichlet domain: Voronoi cell of the orbit of a point x0 with Γx0 = 1. −1 γ2 x0 −1 γ3 x0 −1 γ1 x0 x0 γ1x0 γ x γ3x0 2 0 The points with nontrivial stabilizer are on the boundary An introduction to orbifolds – p.10/20 Structures on orbifolds • Can define geometric structures on orbifolds by taking equivariant definitions on charts Ui, Γi. e • A Riemanian metric on the charts (Ui, Γi) requires: i − U has a Riemannian metric e − coordinatee changes are isometries − Γi acts isometrically on Ui e • e.g. Analytic, flat, hyperbolic, etc... An introduction to orbifolds – p.11/20 Structures on orbifolds • Can define geometric structures on orbifolds by taking equivariant definitions on charts Ui, Γi. e • A Riemanian metric on the charts (Ui, Γi) requires: i − U has a Riemannian metric e − coordinatee changes are isometries − Γi acts isometrically on Ui e • e.g. Analytic, flat, hyperbolic, etc... 2 2 4 2 2 2 6 has a flat metric (Euclidean) has a metric of curv −1 (hyperbolic) An introduction to orbifolds – p.11/20 Structures on orbifolds • Can define geometric structures on orbifolds by taking equivariant definitions on charts Ui, Γi. 2 e 4 2 4 2 4 has a flat metric (Euclidean) has a metric of curv −1 (hyperbolic) An introduction to orbifolds – p.11/20 Coverings p : O0 → O1 is an orbifold covering if −1 Every x ∈ O1 is in some U ⊂ O1 s.t.