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CR Structures Of Once-Punctured Torus Bundles
Alex Casella——————————————————————————
Monash Topology Seminar——————————————————————————
16th May 2018
Alex Casella CR Structures Of Torus Bundles 16th May 2018 1 / 9
§.0 Geometry
§.0 Geometry
Topology : understand 3–manifolds ∼ classification
Different approaches: purely topological / algebraic / geometric / ...
A geometry is a pair (G,X) of a topological space X, and a group Gacting on it.
A geometric (G,X)–structure on a 3–manifold M is
dev : M̃→ X hol : π1(M)→ G
hol(γ) · dev(x) = dev(γ · x), γ ∈ π1(M), x ∈ M̃.
Alex Casella CR Structures Of Torus Bundles 16th May 2018 2 / 9
§.1 Once-punctured torus bundles
§.1 Once-punctured torus bundles
Alex Casella CR Structures Of Torus Bundles 16th May 2018 3 / 9
§.1 Once-punctured torus bundles
§.1 Once-punctured torus bundles
Alex Casella CR Structures Of Torus Bundles 16th May 2018 3 / 9
§.1 Once-punctured torus bundles
§.1 Once-punctured torus bundles
Alex Casella CR Structures Of Torus Bundles 16th May 2018 3 / 9
§.1 Once-punctured torus bundles
Layering
Alex Casella CR Structures Of Torus Bundles 16th May 2018 4 / 9
§.1 Once-punctured torus bundles
Layering
Alex Casella CR Structures Of Torus Bundles 16th May 2018 4 / 9
§.1 Once-punctured torus bundles
Layering
Alex Casella CR Structures Of Torus Bundles 16th May 2018 4 / 9
§.2 CR Space (S3, PU(2, 1))
§.2 CR Space (S3,PU(2, 1)): spherical model
Let U(2, 1) be the matrix group preserving the Hermitian form
〈z,w〉 := wTJz, where J =
0 0 10 1 01 0 0
.
Let π : C3 \ {0} → CP2 and
V− := {z ∈ C3 \ {0} | 〈z, z〉 < 0},V0 := {z ∈ C3 \ {0} | 〈z, z〉 = 0}.
Then π(V−) = H2C and
∂H2C = π(V0)
∼= S3 x PU(2, 1) = U(2, 1)/λI
by biholomorphic transformations.
Alex Casella CR Structures Of Torus Bundles 16th May 2018 5 / 9
§.2 CR Space (S3, PU(2, 1))
§.2 CR Space (S3,PU(2, 1)): spherical model
Let U(2, 1) be the matrix group preserving the Hermitian form
〈z,w〉 := wTJz, where J =
0 0 10 1 01 0 0
.
Let π : C3 \ {0} → CP2 and
V− := {z ∈ C3 \ {0} | 〈z, z〉 < 0},V0 := {z ∈ C3 \ {0} | 〈z, z〉 = 0}.
Then π(V−) = H2C and
∂H2C = π(V0)∼= S3 x PU(2, 1) = U(2, 1)/λI
by biholomorphic transformations.
Alex Casella CR Structures Of Torus Bundles 16th May 2018 5 / 9
§.2 CR Space (S3, PU(2, 1))
§.2 CR Space (S3,PU(2, 1)): Heisenberg modelLet U(2, 1) be the matrix group preserving the Hermitian form
〈z,w〉 := wTJz, where J =
(0 0 10 1 01 0 0
).
Let π : C3 \ {0} → CP2 and
V− := {z ∈ C3 \ {0} | 〈z, z〉 < 0},V0 := {z ∈ C3 \ {0} | 〈z, z〉 = 0}.
Then π(V−) = H2C and
∂H2C = π(V0)∼= S3 x PU(2, 1) = U(2, 1)/λI
by biholomorphic transformations.
Heisenberg Model
The Heisenberg group is H = C× R with the group law
(z1, t1) · (z2, t2) := (z1 + z2, t1 + t2 + 2=(z1z2)).
By stereographic projection Λ : ∂H2C → H = H ∪ {∞}.
Alex Casella CR Structures Of Torus Bundles 16th May 2018 6 / 9
§.2 CR Space (S3, PU(2, 1))
The standard symmetric tetrahedron
Vertices: P1 = [1,√
3],P2 = [1+i√
32 ,√
3],P3 = [0, 0],P4 =∞;
Edges: [P3,P1], [P3,P2], [P3,P4], [P1,P4], [P2,P4], [P1,P2];
Faces: [P4,P3,P1], [P4,P3,P2], [P4,P1,P2], [P2,P3,P1];
Alex Casella CR Structures Of Torus Bundles 16th May 2018 7 / 9
§.2 CR Space (S3, PU(2, 1))
The standard symmetric tetrahedron
Vertices: P1 = [1,√
3],P2 = [1+i√
32 ,√
3],P3 = [0, 0],P4 =∞;
Edges: [P3,P1], [P3,P2], [P3,P4], [P1,P4], [P2,P4], [P1,P2];
Faces: [P4,P3,P1], [P4,P3,P2], [P4,P1,P2], [P2,P3,P1];
Alex Casella CR Structures Of Torus Bundles 16th May 2018 7 / 9
§.2 CR Space (S3, PU(2, 1))
The standard symmetric tetrahedron
Vertices: P1 = [1,√
3],P2 = [1+i√
32 ,√
3],P3 = [0, 0],P4 =∞;
Edges: [P3,P1], [P3,P2], [P3,P4], [P1,P4], [P2,P4], [P1,P2];
Faces: [P4,P3,P1], [P4,P3,P2], [P4,P1,P2], [P2,P3,P1];
Alex Casella CR Structures Of Torus Bundles 16th May 2018 7 / 9
§.3 CR structures on once-punctured torus bundles
§.3 CR structures on once-punctured torus bundles
Theorem
Almost every once-punctured torus bundle admits a branched CRstructure.
1 Monodromy ideal triangulation ∼ new cell decomposition;2 Realise building blocks geometrically;3 Realise face pairings as elements of PU(2, 1);4 Check the edges.
Alex Casella CR Structures Of Torus Bundles 16th May 2018 8 / 9
§.3 CR structures on once-punctured torus bundles
§.3 CR structures on once-punctured torus bundles
Theorem
Almost every once-punctured torus bundle admits a branched CRstructure.
1 Monodromy ideal triangulation ∼ new cell decomposition;
2 Realise building blocks geometrically;3 Realise face pairings as elements of PU(2, 1);4 Check the edges.
Alex Casella CR Structures Of Torus Bundles 16th May 2018 8 / 9
§.3 CR structures on once-punctured torus bundles
§.3 CR structures on once-punctured torus bundles
Theorem
Almost every once-punctured torus bundle admits a branched CRstructure.
1 Monodromy ideal triangulation ∼ new cell decomposition;2 Realise building blocks geometrically;
3 Realise face pairings as elements of PU(2, 1);4 Check the edges.
Alex Casella CR Structures Of Torus Bundles 16th May 2018 8 / 9
§.3 CR structures on once-punctured torus bundles
§.3 CR structures on once-punctured torus bundles
Theorem
Almost every once-punctured torus bundle admits a branched CRstructure.
1 Monodromy ideal triangulation ∼ new cell decomposition;2 Realise building blocks geometrically;3 Realise face pairings as elements of PU(2, 1);
4 Check the edges.
Alex Casella CR Structures Of Torus Bundles 16th May 2018 8 / 9
§.3 CR structures on once-punctured torus bundles
§.3 CR structures on once-punctured torus bundles
Theorem
Almost every once-punctured torus bundle admits a branched CRstructure.
1 Monodromy ideal triangulation ∼ new cell decomposition;2 Realise building blocks geometrically;3 Realise face pairings as elements of PU(2, 1);4 Check the edges.
Alex Casella CR Structures Of Torus Bundles 16th May 2018 8 / 9
§.3 CR structures on once-punctured torus bundles
Thank You!
Thank you very much for your attention!
Alex Casella CR Structures Of Torus Bundles 16th May 2018 9 / 9
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