Spherical three-dimensional orbifolds

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Spherical three-dimensional orbifolds Andrea Seppi joint work with Mattia Mecchia Pisa, 16th May 2013

Outline What is an orbifold? What is a spherical orientable orbifold? What is a fibered orbifold? An algebraic classification A geometric classification How to compare the two classifications

Outline What is an orbifold? What is a spherical orientable orbifold? What is a fibered orbifold? An algebraic classification A geometric classification How to compare the two classifications

Outline What is an orbifold? What is a spherical orientable orbifold? What is a fibered orbifold? An algebraic classification A geometric classification How to compare the two classifications

Outline What is an orbifold? What is a spherical orientable orbifold? What is a fibered orbifold? An algebraic classification A geometric classification How to compare the two classifications

Outline What is an orbifold? What is a spherical orientable orbifold? What is a fibered orbifold? An algebraic classification A geometric classification How to compare the two classifications

Outline What is an orbifold? What is a spherical orientable orbifold? What is a fibered orbifold? An algebraic classification A geometric classification How to compare the two classifications

What is an orbifold? Definition A smooth orbifold of dimension n is a Hausdorff paracompact topological space O provided with a collection of homeomorphisms (charts) ffi i : Ũ i =Γ i! U i Ũ i are connected open sets in R n Γ i is a finite group acting effectively and smoothly on Ũ i The U i s cover O Charts are compatible

Compatibility of charts Ũ i =Γ i ı i ffi i U i \ U j ffi j Ũ j =Γ j ij Ṽ i Ũ i Ṽ j Ũ j ı j If U i \ U j 6= ;, we require coordinate changes to be locally lifted by a diffeomorphism ij : Ṽ i! Ṽ j. Riemannian Orbifold: Ũ i are provided with Riemannian metrics preserved by Γ i and ij Spherical Orbifold: Ũ i are provided with Riemannian metrics of constant curvature +1 preserved by Γ i e ij Orientable Orbifold: Ũ i are provided with an orientation preserved by Γ i and ij

Compatibility of charts Ũ i =Γ i ı i ffi i U i \ U j ffi j Ũ j =Γ j ij Ṽ i Ũ i Ṽ j Ũ j ı j If U i \ U j 6= ;, we require coordinate changes to be locally lifted by a diffeomorphism ij : Ṽ i! Ṽ j. Riemannian Orbifold: Ũ i are provided with Riemannian metrics preserved by Γ i and ij Spherical Orbifold: Ũ i are provided with Riemannian metrics of constant curvature +1 preserved by Γ i e ij Orientable Orbifold: Ũ i are provided with an orientation preserved by Γ i and ij

Compatibility of charts Ũ i =Γ i ı i ffi i U i \ U j ffi j Ũ j =Γ j ij Ṽ i Ũ i Ṽ j Ũ j ı j If U i \ U j 6= ;, we require coordinate changes to be locally lifted by a diffeomorphism ij : Ṽ i! Ṽ j. Riemannian Orbifold: Ũ i are provided with Riemannian metrics preserved by Γ i and ij Spherical Orbifold: Ũ i are provided with Riemannian metrics of constant curvature +1 preserved by Γ i e ij Orientable Orbifold: Ũ i are provided with an orientation preserved by Γ i and ij

Compatibility of charts Ũ i =Γ i ı i ffi i U i \ U j ffi j Ũ j =Γ j ij Ṽ i Ũ i Ṽ j Ũ j ı j If U i \ U j 6= ;, we require coordinate changes to be locally lifted by a diffeomorphism ij : Ṽ i! Ṽ j. Riemannian Orbifold: Ũ i are provided with Riemannian metrics preserved by Γ i and ij Spherical Orbifold: Ũ i are provided with Riemannian metrics of constant curvature +1 preserved by Γ i e ij Orientable Orbifold: Ũ i are provided with an orientation preserved by Γ i and ij

Some examples G is a cyclic group acting on S 2, generated by a rotation of 2ı n : The quotient S 2 =G has underlying topological space S 2 and two singular points with group Z n (cone points).

Some examples G is a dihedral group acting on S 2, generated by two reflections in planes which form an angle of ı n : The quotient S 2 =G has underlying topological space D 2 and singular points on the boundary with group D 2n (corner reflectors) or Z 2 (mirror reflectors).

Theorem Every compact spherical orbifold is diffeomorphic to S n =Γ, where Γ is a finite subgroup of O(n + 1). Theorem Every orientable spherical orbifold is diffeomorphic to S n =Γ, where Γ is a finite subgroup of SO(n + 1). Theorem Two compact spherical orbifolds S n =Γ and S n =Λ are isometric if and only if Γ and Λ are conjugated in O(n + 1). Theorem Two compact orientable spherical orbifolds S n =Γ and S n =Λ are orientation-preserving isometric if and only if Γ and Λ are conjugated in SO(n + 1).

Theorem Every compact spherical orbifold is diffeomorphic to S n =Γ, where Γ is a finite subgroup of O(n + 1). Theorem Every orientable spherical orbifold is diffeomorphic to S n =Γ, where Γ is a finite subgroup of SO(n + 1). Theorem Two compact spherical orbifolds S n =Γ and S n =Λ are isometric if and only if Γ and Λ are conjugated in O(n + 1). Theorem Two compact orientable spherical orbifolds S n =Γ and S n =Λ are orientation-preserving isometric if and only if Γ and Λ are conjugated in SO(n + 1).

Theorem Every compact spherical orbifold is diffeomorphic to S n =Γ, where Γ is a finite subgroup of O(n + 1). Theorem Every orientable spherical orbifold is diffeomorphic to S n =Γ, where Γ is a finite subgroup of SO(n + 1). Theorem Two compact spherical orbifolds S n =Γ and S n =Λ are isometric if and only if Γ and Λ are conjugated in O(n + 1). Theorem Two compact orientable spherical orbifolds S n =Γ and S n =Λ are orientation-preserving isometric if and only if Γ and Λ are conjugated in SO(n + 1).

Theorem Every compact spherical orbifold is diffeomorphic to S n =Γ, where Γ is a finite subgroup of O(n + 1). Theorem Every orientable spherical orbifold is diffeomorphic to S n =Γ, where Γ is a finite subgroup of SO(n + 1). Theorem Two compact spherical orbifolds S n =Γ and S n =Λ are isometric if and only if Γ and Λ are conjugated in O(n + 1). Theorem Two compact orientable spherical orbifolds S n =Γ and S n =Λ are orientation-preserving isometric if and only if Γ and Λ are conjugated in SO(n + 1).

The bijection algebraic description Finite subgroups Γ of SO(4) up to conjugation in SO(4), geometric description Spherical orientable orbifolds S 3 =Γ in the quotient up to orientation-preserving isometries

The algebraic side We need a classification of finite subgroups of SO(4) up to conjugation. Patrick DuVal Homographies, quaternions and rotations Oxford Mathematical Monographies, 1964

Real quaternion algebra defines an algebraic structure on S 3. S 3 = (a; b; c; d) 2 R 4 ; a 2 + b 2 + c 2 + d 2 = 1 = (z 1 ; z 2 ) : z 1 ; z 2 2 C; jz 1 j 2 + jz 2 j 2 = 1 = z 1 + z 2 j : z 1 ; z 2 2 C; jz 1 j 2 + jz 2 j 2 = 1 Define a homomorphism : S 3 ˆ S 3! SO(4) (p; p 0 ) : q 7! pq(p 0 )`1 is surjective and its kernel is ker = f(1; 1); (`1; `1)g Then we use to classify finite subgroups G SO(4) by their preimage G 0 = `1 (G) S 3 ˆ S 3.

The geometric side On the other hand, we want to give a geometric description of S 3 =Γ up to isometries. In most cases, these are described by a Seifert fibration. William Dunbar Fibered orbifolds and crystallographic groups PhD Thesis, 1981 Francis Bonahon and Laurent Siebenmann The classification of Seifert fibred 3-orbifolds Low-dimensional topology, 1985

Definition A Seifert fibration for orbifolds is given by an orbifold O (dimo = 3), a base orbifold B (dimb = 2) and a smooth surjective projection p : O! B, such that for every x 2 B the following diagram commutes: p`1 (U) (Ũ ˆ S 1 )=Γ x Ũ ˆ S 1 p U = Ũ=Γ x q ı Ũ pr 1 Theorem A spherical orientable orbifold S 3 =Γ admits a Seifert fibration if and only if Γ SO(4) preserves a Seifert fibration of S 3.

Definition A Seifert fibration for orbifolds is given by an orbifold O (dimo = 3), a base orbifold B (dimb = 2) and a smooth surjective projection p : O! B, such that for every x 2 B the following diagram commutes: p`1 (U) (Ũ ˆ S 1 )=Γ x Ũ ˆ S 1 p U = Ũ=Γ x q ı Ũ pr 1 Theorem A spherical orientable orbifold S 3 =Γ admits a Seifert fibration if and only if Γ SO(4) preserves a Seifert fibration of S 3.

The bijection: the fibered case algebraic classification Finite subgroups Γ of SO(4) Γ preserves a Seifert fibration of S 3, geometric classification Spherical orientable orbifolds S 3 =Γ in the quotient S 3 =Γ inherits a Seifert fibration

Stereographic projection In dimension two,we can map S 2 n (0; 0; 1) to R 2 and S 2 to R 2 [ f1g, by stereographic projection. In the same way, let us visualize S 3 as R 3 [ f1g, under stereographic projection.

The Hopf fibration S 3 = (z 1 ; z 2 ) : z 1 ; z 2 2 C; jz 1 j 2 + jz 2 j 2 = 1 is the union of two solid tori T 1 = jz 1 j 2» 1 2 and T2 = jz 1 j 2 1 2. Hopf ``!

The family of subgroups (C 2mr =C m ; C 2nr =C n ) s The first family of subgroups G in the list is (C 2mr =C m ; C 2nr =C n ) s This means that, if G 0 = `1 (G) S 3 ˆ S 3 C 2mr = ı 1 (G 0 ) = Z 2mr C 2nr = ı 2 (G 0 ) = Z 2nr C m = ı 1 (G 0 \ (S 3 ˆ f1g)) = Z m C n = ı 2 (G 0 \ (f1g ˆ S 3 )) = Z n and there is an isomorphism ffi : C 2mr =C m = Zr! C 2nr =C n = Zr ffi(1) = s

Singularities = mcd(n + sm; n ` sm; mnr ) 1 = mcd( n+sm ; mnr ), 2 = mcd( n`sm ; mnr ) G contains a cyclic subgroup Z 1 which fixes the core of the solid torus T 1 pointwise. In the quotient,we have a singular fiber of order 1. Analogoulsy we have a singular fiber of order 2 due to the action of a cyclic subgroup Z 2 fixing the core of T 2 pointwise.

Singularities = mcd(n + sm; n ` sm; mnr ) 1 = mcd( n+sm ; mnr ), 2 = mcd( n`sm ; mnr ) G contains a cyclic subgroup Z 1 which fixes the core of the solid torus T 1 pointwise. In the quotient,we have a singular fiber of order 1. Analogoulsy we have a singular fiber of order 2 due to the action of a cyclic subgroup Z 2 fixing the core of T 2 pointwise.

Singularities = mcd(n + sm; n ` sm; mnr ) 1 = mcd( n+sm ; mnr ), 2 = mcd( n`sm ; mnr ) G contains a cyclic subgroup Z 1 which fixes the core of the solid torus T 1 pointwise. In the quotient,we have a singular fiber of order 1. Analogoulsy we have a singular fiber of order 2 due to the action of a cyclic subgroup Z 2 fixing the core of T 2 pointwise.

The underlying manifold Other elements are acting with no fixed points, by a composite rotation. They still keep T 1 and T 2 invariant. The quotient orbifold is the union of two solid tori by a nontrivial glueing on their boundary. The underlying topological space is a lens space L( ; `1 ), where: = mnr, = j2 (n+sm)+2mnr, = j2 (n`sm)`2mnr. 2 1 2 2 j 1 2 j 2

The underlying manifold Other elements are acting with no fixed points, by a composite rotation. They still keep T 1 and T 2 invariant. The quotient orbifold is the union of two solid tori by a nontrivial glueing on their boundary. The underlying topological space is a lens space L( ; `1 ), where: = mnr, = j2 (n+sm)+2mnr, = j2 (n`sm)`2mnr. 2 1 2 2 j 1 2 j 2

The underlying manifold Other elements are acting with no fixed points, by a composite rotation. They still keep T 1 and T 2 invariant. The quotient orbifold is the union of two solid tori by a nontrivial glueing on their boundary. The underlying topological space is a lens space L( ; `1 ), where: = mnr, = j2 (n+sm)+2mnr, = j2 (n`sm)`2mnr. 2 1 2 2 j 1 2 j 2

The Seifert fibration We can describe the quotient orbifold S 3 =G as a Seifert fibered orbifold: Base orbifold Euler number e = m nr Local invariants (js + 2n j r )`1 nr 2 1 ` (js + 2n j r )`1 nr 2 2

The family of subgroups (D 2mr =C m ; D 2nr =C n ) s Other elements are acting on each torus T 1 and T 2 by hyperelliptic involution. The quotient orbifold is the union of two 3-balls by a glueing on their boundary. The underlying topological space is S 3.

The family of subgroups (D 2mr =C m ; D 2nr =C n ) s Other elements are acting on each torus T 1 and T 2 by hyperelliptic involution. The quotient orbifold is the union of two 3-balls by a glueing on their boundary. The underlying topological space is S 3.

The family of subgroups (D 2mr =C m ; D 2nr =C n ) s Other elements are acting on each torus T 1 and T 2 by hyperelliptic involution. The quotient orbifold is the union of two 3-balls by a glueing on their boundary. The underlying topological space is S 3.

The family of subgroups (D 2mr =C m ; D 2nr =C n ) s Other elements are acting on each torus T 1 and T 2 by hyperelliptic involution. The quotient orbifold is the union of two 3-balls by a glueing on their boundary. The underlying topological space is S 3.

The Seifert fibration We can describe the quotient orbifold S 3 =G as a Seifert fibered orbifold: Base orbifold Euler number e = m 2nr Local invariants (js + 2n j r )`1 nr 2 1 ` (js + 2n j r )`1 nr 2 2

Again S 3 Decompose the base orbifold by cutting out a disc. The preimage of the disc is a solid torus with the trivial fibration The union of the two solid tori is S 3!

Again S 3 Decompose the base orbifold by cutting out a disc. The preimage of the disc is a solid torus with the trivial fibration The union of the two solid tori is S 3!

Again S 3 Decompose the base orbifold by cutting out a disc. The preimage of the disc is a solid torus with the trivial fibration The union of the two solid tori is S 3!

Again S 3 Decompose the base orbifold by cutting out a disc. The preimage of the disc is a solid torus with the trivial fibration The union of the two solid tori is S 3!

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