Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds ORBIFOLDS July 16, 2010 ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Outline 1 Definition and Examples of Orbifolds 2 Orbifolds as quotients of manifolds by Lie group actions 3 Stratification of orbifolds ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Outline 1 Definition and Examples of Orbifolds 2 Orbifolds as quotients of manifolds by Lie group actions 3 Stratification of orbifolds ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds What is an orbifold? Definition An n-dimensional orbifold O is a second countable Hausdorff topological space together with a maximal n-dimensional orbifold atlas. Definition An orbifold atlas is a collection of mutually compatible orbifold charts whose images cover O. So what’s an orbifold chart? ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds What is an orbifold? Definition An n-dimensional orbifold O is a second countable Hausdorff topological space together with a maximal n-dimensional orbifold atlas. Definition An orbifold atlas is a collection of mutually compatible orbifold charts whose images cover O. So what’s an orbifold chart? ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds What is an orbifold? Definition An n-dimensional orbifold O is a second countable Hausdorff topological space together with a maximal n-dimensional orbifold atlas. Definition An orbifold atlas is a collection of mutually compatible orbifold charts whose images cover O. So what’s an orbifold chart? ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Orbifold charts Definition An orbifold chart or uniformizing system (Ue, ΓU , πU ) on a connected open set U is given by ∼ / / Γ \U / U Ue U e = πu where Ue is a connected open subset of Rn ΓU is a finite group acting on Ue by diffeomorphisms πU : Ue → U is a continuous map inducing a homeomorphism from the orbit space ΓU \Ue onto U. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Orbifold charts Definition An orbifold chart or uniformizing system (Ue, ΓU , πU ) on a connected open set U is given by ∼ / / Γ \U / U Ue U e = πu where Ue is a connected open subset of Rn ΓU is a finite group acting on Ue by diffeomorphisms πU : Ue → U is a continuous map inducing a homeomorphism from the orbit space ΓU \Ue onto U. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Orbifold charts Definition An orbifold chart or uniformizing system (Ue, ΓU , πU ) on a connected open set U is given by ∼ / / Γ \U / U Ue U e = πu where Ue is a connected open subset of Rn ΓU is a finite group acting on Ue by diffeomorphisms πU : Ue → U is a continuous map inducing a homeomorphism from the orbit space ΓU \Ue onto U. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Orbifold charts Definition An orbifold chart or uniformizing system (Ue, ΓU , πU ) on a connected open set U is given by ∼ / / Γ \U / U Ue U e = πu where Ue is a connected open subset of Rn ΓU is a finite group acting on Ue by diffeomorphisms πU : Ue → U is a continuous map inducing a homeomorphism from the orbit space ΓU \Ue onto U. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Orbifold charts Definition An orbifold chart or uniformizing system (Ue, ΓU , πU ) on a connected open set U is given by ∼ / / Γ \U / U Ue U e = πu where Ue is a connected open subset of Rn ΓU is a finite group acting on Ue by diffeomorphisms πU : Ue → U is a continuous map inducing a homeomorphism from the orbit space ΓU \Ue onto U. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Orbifold charts Definition An orbifold chart or uniformizing system (Ue, ΓU , πU ) on a connected open set U is given by ∼ / / Γ \U / U Ue U e = πu where Ue is a connected open subset of Rn ΓU is a finite group acting on Ue by diffeomorphisms πU : Ue → U is a continuous map inducing a homeomorphism from the orbit space ΓU \Ue onto U. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Isomorphic charts Definition 0 An isomorphism between two charts (Ue, ΓU , πU ) and (Ue , ΓU0 , πU0 ) on the same open set U is a diffeomorphism φ : Ue → Uf0 such that πU0 ◦ φ = φ ◦ πU . ϕ U / U0 e 3 ' f 33 π 3 π U 3 U0 3 Ô U Necessarily ΓU ' ΓU0 and ϕ is equivariant. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Isomorphic charts Definition 0 An isomorphism between two charts (Ue, ΓU , πU ) and (Ue , ΓU0 , πU0 ) on the same open set U is a diffeomorphism φ : Ue → Uf0 such that πU0 ◦ φ = φ ◦ πU . ϕ U / U0 e 3 ' f 33 π 3 π U 3 U0 3 Ô U Necessarily ΓU ' ΓU0 and ϕ is equivariant. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Automorphisms of charts Proposition Every automorphism (φ, ΓU ) of a chart (Ue, ΓU , πU ) is inner. I.e., ϕ ∈ ΓU . ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Automorphisms of charts Proposition Every automorphism (φ, ΓU ) of a chart (Ue, ΓU , πU ) is inner. I.e., ϕ ∈ ΓU . ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Automorphisms of charts Proposition Every automorphism (φ, ΓU ) of a chart (Ue, ΓU , πU ) is inner. I.e., ϕ ∈ ΓU . ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Injections of charts Definition An injection (Ve, ΓV , πV ) → (Ue, ΓU , πU ) of charts on V ⊆ U is an open embedding ϕ : Ve → Ue such that the diagram ϕ Ve / / Ue πV πU V / U commutes. Given such an injection, there exists a monomorphism λ :ΓV → ΓU such that ϕ ◦ γ = λ(γ) ◦ ϕ for all γ ∈ ΓU . ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Injections of charts Definition An injection (Ve, ΓV , πV ) → (Ue, ΓU , πU ) of charts on V ⊆ U is an open embedding ϕ : Ve → Ue such that the diagram ϕ Ve / / Ue πV πU V / U commutes. Given such an injection, there exists a monomorphism λ :ΓV → ΓU such that ϕ ◦ γ = λ(γ) ◦ ϕ for all γ ∈ ΓU . ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Induced charts Proposition Given a chart (Ue, ΓU , πU ) on an open set U and given a connected open subset W ⊂ U, there exists a unique up to isomorphism chart (Wf, ΓW , πW ) on W that injects into (Ue, ΓU , πU ). This is the chart induced on W by (Ue, ΓU , πU ). ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Induced charts Proposition Given a chart (Ue, ΓU , πU ) on an open set U and given a connected open subset W ⊂ U, there exists a unique up to isomorphism chart (Wf, ΓW , πW ) on W that injects into (Ue, ΓU , πU ). This is the chart induced on W by (Ue, ΓU , πU ). ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Compatibility of charts Definition 0 0 Two charts (Ue, ΓU , πU ) and (Ue , ΓU0 , πU0 ) on open subsets U and U are said to be compatible if for every x ∈ U ∩ U0, there exists a neighborhood x ∈ W ⊂ U ∩ U0 0 such that (Ue, ΓU , πU ) and (Ue , ΓU0 , πU0 ) induce isomorphic charts on W . Caution: There’s a subtlety here. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds Compatibility of charts Definition 0 0 Two charts (Ue, ΓU , πU ) and (Ue , ΓU0 , πU0 ) on open subsets U and U are said to be compatible if for every x ∈ U ∩ U0, there exists a neighborhood x ∈ W ⊂ U ∩ U0 0 such that (Ue, ΓU , πU ) and (Ue , ΓU0 , πU0 ) induce isomorphic charts on W . Caution: There’s a subtlety here. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds What is an orbifold? Definition An orbifold atlas is a collection of mutually compatible orbifold charts whose images cover O. Definition An n-dimensional orbifold O is a second countable Hausdorff topological space together with a maximal n-dimensional orbifold atlas. ORBIFOLDS Definition and Examples of Orbifolds Orbifolds as quotients of manifolds by Lie group actions Stratification of orbifolds What is an orbifold? Definition An orbifold atlas is a collection of mutually compatible orbifold charts whose images cover O.