Fundamental Groupoids for Orbifolds Laura Scull joint with Dorette - PowerPoint PPT Presentation

Fundamental Groupoids for Orbifolds Laura Scull joint with Dorette Pronk and Courtney Thatcher

Orbifolds An orbifold is: • a generalization of a manifold • a space that is locally modelled by quotients of R n by actions of finite groups • allows controlled singularities

Example 0: A Manifold (with boundary)

Example 1: A Cone Point

Example 2: Silvered Interval

Example 3: Mirrored Boundary Disk

Example 4: The Teardrop

Example 5: The Billiard Table

Example 6: Ineffective Z/3 Action Z/3 Isotropy

Orbifolds via Atlases (Effective Edition) We can represent an orbifold using charts making up an atlas: • � U is a connected open subset of R n ; • G is a finite group acting effectively on � U ; • π : � U → U is a continuous and surjective map that induces a homeomorphism between U and � U / G

Orbifolds via Atlases (Effective Edition) Charts creating an atlas: • A collection of charts U such that the quotients cover the underlying space, and all chart embeddings between them. • The charts are required to be locally compatible: for any two charts for subsets U , V ⊆ X and any point x ∈ U ∩ V , there is a neighbourhood W ⊆ U ∩ V containing x with a chart ( � W , G W , π W ) in U , and chart embeddings into ( � U , G U , π U ) and ( � V , G V , π V ) .

Orbifolds via Atlases (Effective Edition) For any µ ji in O ( U ) , the set Emb ( µ ji ) forms an atlas bimodule Emb ( µ ji ) : G i � −→ G j . with actions given by composition. If i = j the atlas bimodule Emb ( µ ii ) is isomorphic to the trivial bimodule G i associated to the group G i . Furthermore, these define a pseudofunctor Emb : O ( U ) −→ GroupMod , with Emb ( U i ) := G i on objects, and Emb ( µ ji ) : G i � −→ G j on morphisms.

Orbifolds via Atlases (General Edition) Let U be a non-empty connected topological space; an orbifold chart (also known as a uniformizing system ) of dimension n for U is a quadruple ( � U , G , ρ, π ) where: • � U is a connected and simply connected open subset of R n ; • G is a finite group; • ρ : G → Aut ( � U ) is a (not necessarily faithful) representation of G as a group of smooth automorphisms of � U ; we set G red := ρ ( G ) ⊆ Aut ( � U ) and Ker ( G ) := Ker ( ρ ) ⊆ G ; • π : � U → U is a continuous and surjective map that induces a homeomorphism between U and � U / G red .

Orbifolds via Atlases (General Edition) An orbifold atlas of dimension n for X is: 1. a collection U = { ( � U i , G i , ρ i , π i ) } i ∈ I of orbifold charts, of dimension n , connected and simply connected, such that the reduced charts { ( � U i , G red , π i ) } i ∈ I form a Satake atlas for i X ; let ( Con , γ ): O ( U ) → GroupMod be the induced pseudofunctor 2. a pseudofunctor Abst : O ( U ) −→ GroupMod such that for each i ∈ I , Abst ( U i ) = G i and for each µ ji in O ( U ) , Abst ( µ ji ) is an atlas bimodule G i � −→ G j , (i.e., the left action of G j is free and transitive and the right action of G i is free).

Orbifolds via Atlases (General Edition) (3) an oplax transformation ρ = ( { ρ ρ i } i ∈ I , { ρ ji } i , j ∈ I , U i ⊆ U j ): Abst ⇒ Con: each ρ i is a group ρ ρ ρ homomorphism from G i to G red , hence it induces a i −→ G red bimodule ρ ρ i : G i � forming the components of the ρ i transformation. We further require that: • the ρ ji are surjective maps of bimodules; • (transitivity on the kernel) whenever ⊗ λ ′ ) for λ, λ ′ ∈ Abst ( µ ji ) , there is an ρ ji ( e red ⊗ λ ) = ρ ji ( e red j j element g ∈ G i such that λ · g = λ ′ (here e red is the identity j element of G red ). j

Orbifolds via G -spaces We can represent some (most?) orbifolds via group actions • the orbifold is the quotient space of a (compact Lie) group acting on a manifold • if the group is finite, the orbifold is a global quotient • unknown whether all orbifolds are representable this way

Orbifolds via Topological Groupoids • A topological groupoid has a space of object G 0 and a space of arrows G 1 , where all structure maps are continuous • G is étale when s (and hence t ) is a local homeomorphism • G is proper when the diagonal, ( s , t ): G 1 → G 0 × G 0 , is a proper map (i.e., closed with compact fibers).

� Orbigroupoids Definition • A topological groupoid is an orbigroupoid if it is both étale and proper. • All isotropy groups are finite. • The quotient space, s � � X G G 1 � G 0 t is also called the underlying space of the orbigroupoid. • This space is an orbifold.

Example 1: A Cone Point

Example 1: A Cone Point as an atlas (with one chart) Z/3

Example 1: A Cone Point as a G -space Z/3

Example 1: A Cone Point as a groupoid 2/3 1/3 arrows e objects

Example 2: Silvered Interval

Example 2: Silvered Interval as an atlas e Z/2 Z/2

Example 2: Silvered Interval as a G -space Z/2

Example 2: Silvered Interval as a groupoid flip arrows e objects

Example 3: Mirrored Boundary Disk

Example 3: Mirrored Boundary Disk as a G -space Z/2

Example 3: Mirrored Boundary Disk as a groupoid flip e arrows objects

Example 4: The Teardrop

Example 4: The Teardrop as an atlas Z/3 e

Example 4: The Teardrop as a groupoid 2/3 1/3 e wrap 3X

Example 5: The Billiard Table

Example 5: The Billiard Table Z/2 D2 D 2 Z/2 Z/2 D2 D 2 Z/2

Example 6: Ineffective Z/3 Action

Example 6: Ineffective Z/3 Action U 1 U 3 U 4 U 2 G U i = Z / 3 and G red = { e } . Forr each inclusion µ ji : U i ֒ → U j , we U i need a module M ji and a map of bimodules ρ ji as follows: Abst ( µ ji )= M ji Z / 3 = { e , ω i , ω 2 Z / 3 = { e , ω j , ω 2 i } j } / ⇓ ρ ji ρ ρ ρ ρ i ρ ρ j / / { e } { e } / Con ( µ ji )= { λ ji } (1)

Example 6: Ineffective Z/3 Action a ji c ji left multiply by ω j b ji a ji c ji b ji right multiply by ω i

Example 6: Ineffective Z/3 Action M 13 , M 14 and M 23 as before, M 24 with action given by a ji c ji b ji left multiply by ω j a ji c ji b ji right multiply by ω i

(Borel) Fundamental Group If G is a groupoid representing an orbifold, we can define a fundamental group by: • π 1 ( B G ) • Haefliger paths • deck transformation of universal cover • homotopy classes of maps I → G

Haefliger paths Let G be a Lie groupoid. A path from x to y in G 0 is: • a subdivision 0 = t 0 < t 1 < t 2 . . . t n = 1 • a sequence ( g 0 , α 1 , g 1 , . . . , α n , g n ) • g i ∈ G 1 such that s ( g 0 ) = x , t ( g n ) = y • α i : [ t i − 1 , t i ] → G 0 is a path from t ( g i − 1 ) to s ( g i ) g g a a g a g a g 1 2 3 2 3 n-1 n n 1 x y

Haefliger Paths Two paths are equivalent if: • we add a new point to the subdivision with an identitiy g i : g = id i g a a i i • we have homotopy h : [ t i − 1 , t i ] → G 1 with s ◦ h i = α i and t ◦ h i = α ′ i and we replace ( . . . g i − 1 , α i , g i , . . . ) by ( . . . h ( t i − 1 ) g i − 1 , α ′ , g i h ( t i ) − 1 , . . . ) g g g a g a i-1 i i i-1 i i a' i

Haefliger Paths Two paths are homotopic if: • we have homotopies h : [ t i − 1 , t i ] × I → G 0 with h ( t , 0 ) = α i and h ( t , 1 ) = α ′ i • we have compatible homotopies K : I → G 1 with K ( 0 ) = g i and K ( 1 ) = g ′ i g g a i-1 i i H K g' g' a' i-1 i i We define the orbifold fundamental groupoid as the homotopy classes of these paths.

Order 3 Cone 2/3 2/3 2/3 1/3 1/3 1/3 e e e

Order 3 Cone 2/3 1/3 e

Order 3 Cone 2/3 2/3 2/3 1/3 1/3 1/3 e e e

Order 3 Cone 2/3 2/3 1/3 1/3 e e

Order 3 Cone 2/3 2/3 1/3 1/3 e e

Order 3 Cone 2/3 2/3 2/3 1/3 1/3 1/3 e e e

Order 3 Cone π 1 ( G ) = Z / 3

Silvered Interval flip flip flip e e e

Silvered Interval π 1 ( G ) = D ∞

Teardrop 2/3 1/3 e wrap 3X

Teardrop 2/3 1/3 e wrap 3X

Teardrop π 1 ( G ) = e

(Borel) Fundamental Group Recall we can define π 1 ( G ) by: • π 1 ( B G ) • Haefliger paths • deck transformation of universal cover • homotopy classes of maps I → G

(Borel) Fundamental Group B G defined by the geometric realization of the nerve of G : • ∆ 0 for every x ∈ G 0 • ∆ 1 for every g ∈ G 1 attached to s ( g ) and t ( g ) g x y • ∆ 2 for every composible ( g 1 , g 2 ) attached by g 1 , g 2 , g 2 g 1 g1 g g g2 2 1 • higher simplices attached but do not affect π 1

(Borel) Fundamental Group π 1 ( B G ) is the Haefliger group • a path in π 1 ( BG ) can follow a line in B G corresponding to g ∈ G 1 , giving a hop • paths can be homotopic over triangles corresponding to equivalence of Haefliger paths

(Borel) Fundamental Group Defined via deck transformations (topos) Defined via homotopy classes of maps I → G :