11.1 Geometric structures on 3-manifolds

Thurston geometries are eight fundamental structures that shape 3-manifolds. They range from flat Euclidean space to twisted Nil geometry, each with unique properties. Understanding these geometries is key to grasping the topology of 3-dimensional spaces.

The geometrization conjecture links these geometric structures to the topology of 3-manifolds. It's a powerful tool that helps classify 3-manifolds and solve long-standing problems like the Poincaré conjecture. This connection between geometry and topology is central to modern 3-manifold theory.

Thurston Geometries for 3-Manifolds

Classification and Characteristics

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• Eight Thurston geometries construct 3-manifolds (Euclidean (E³), Spherical (S³), Hyperbolic (H³), S² × R, H² × R, Nil, Sol, and SL(2,R))
• Each Thurston geometry represents a simply connected homogeneous space with a transitive group of isometries
• Euclidean geometry (E³) exhibits zero curvature and flat space properties
• Spherical geometry (S³) demonstrates positive constant curvature analogous to a three-dimensional sphere surface
• Hyperbolic geometry (H³) showcases negative constant curvature in three-dimensional hyperbolic space
• Product geometries (S² × R and H² × R) combine lower-dimensional geometries with a real line
  • S² × R combines a two-dimensional sphere with a real line
  • H² × R combines a two-dimensional hyperbolic plane with a real line
• Nil geometry bases on the Heisenberg group with a twisted product structure
  • Heisenberg group consists of 3x3 upper triangular matrices with 1s on the diagonal
• Sol geometry derives from isometries of two-dimensional Minkowski space
  • Minkowski space serves as the mathematical foundation for special relativity
• SL(2,R) geometry bases on the universal cover of the special linear group SL(2,R)
  • SL(2,R) consists of 2x2 matrices with determinant 1

Unique Properties and Applications

• Euclidean geometry applies to flat tori and other manifolds with zero curvature
• Spherical geometry describes manifolds like lens spaces and Poincaré homology spheres
• Hyperbolic geometry applies to knot complements and closed surfaces of genus ≥ 2
• S² × R geometry describes manifolds like S² × S¹ (trivial 2-sphere bundle over circle)
• H² × R geometry applies to manifolds like surface bundles over circles with hyperbolic fiber
• Nil geometry describes certain circle bundles over tori
• Sol geometry applies to torus bundles over circles with hyperbolic monodromy
• SL(2,R) geometry describes unit tangent bundles of hyperbolic surfaces

Geometry vs Topology of 3-Manifolds

Fundamental Connections

• Geometrization Conjecture (proven by Perelman) connects geometric structures to topological properties of 3-manifolds
• Poincaré conjecture states every simply connected closed 3-manifold is homeomorphic to the 3-sphere
  • Special case of the Geometrization Conjecture
  • Resolved the last remaining Millennium Prize Problem in mathematics
• Thurston's hyperbolization theorem relates hyperbolic geometry to Haken manifold topology
  • Haken manifolds contain incompressible surfaces
• Volume of hyperbolic 3-manifolds serves as a topological invariant
  • Connects geometric measurement to topological classification
• Dehn filling operations on cusped hyperbolic 3-manifolds produce topologically distinct manifolds
  • Alters geometric structures while changing topology
• Gromov norm of 3-manifolds relates to hyperbolic piece decomposition
  • Bridges geometry and topology through simplicial volume
• Geometric transitions between Thurston geometries occur through deformations
  • Reflects topological changes in the underlying manifold structure

Analytical Tools and Theorems

• JSJ decomposition provides insights into applicable geometric structures for manifold pieces
  • Decomposes 3-manifolds along essential tori and Klein bottles
• Mostow's Rigidity Theorem proves uniqueness of hyperbolic structures on finite-volume hyperbolic 3-manifolds
  • Implies volume is a topological invariant for hyperbolic 3-manifolds
• Seifert-Weber dodecahedral space admits hyperbolic structure
  • Proven using fundamental polyhedron and gluing isometries
• 3-torus admits Euclidean structure
  • Constructed as quotient of E³ by lattice of translations
• Non-existence proofs involve obstructions (Euler characteristic, fundamental group properties, incompatible cusp structures)
  • Euler characteristic obstruction applies to S³ geometry on manifolds with χ ≠ 0
• Dunbar and Meyerhoff's classification of non-compact hyperbolic 3-orbifolds aids in proving (non-)existence of hyperbolic structures
  • Provides comprehensive list of possible non-compact hyperbolic 3-orbifolds

Applying Geometry to 3-Manifolds

Topological Properties and Geometric Structures

• Fundamental group of 3-manifolds crucially determines admissible geometric structures
  • Finite fundamental groups admit spherical geometry (S³)
  • Infinite cyclic fundamental groups compatible with S² × R geometry
• Torus bundles over circle admit various geometries based on monodromy matrix
  • Euclidean geometry for monodromy with trace = 2
  • Nil geometry for monodromy with trace = ±2 (non-identity)
  • Sol geometry for hyperbolic monodromy (|trace| > 2)
• Hyperbolic structures apply to 3-manifolds with fundamental groups as discrete subgroups of PSL(2,C)
  • PSL(2,C) represents the group of orientation-preserving isometries of H³
• Seifert fibered structure often indicates compatibility with specific geometries
  • E³, S³, S² × R, H² × R, or Nil geometry depending on the base orbifold and Euler number

Geometric Transitions and Deformations

• Dehn filling operations alter geometric structures of cusped hyperbolic 3-manifolds
  • Can transition from hyperbolic to other geometries (Euclidean, spherical)
• Cone manifold deformations provide smooth transitions between geometric structures
  • Example: deforming hyperbolic structure to spherical structure through Euclidean
• Collapsing or expanding certain directions in geometric structures leads to transitions
  • Collapsing S³ along Hopf fibration results in S² geometry
• Deforming complex structures on surfaces induces changes in associated SL(2,R) structures
  • Relates to Teichmüller theory and moduli spaces of geometric structures

Existence of Geometric Structures on 3-Manifolds

Proof Techniques and Theorems

• Thurston's Geometrization Theorem provides framework for proving geometric structure existence on prime 3-manifolds
  • Decomposes 3-manifolds into pieces admitting unique geometric structures
• Mostow's Rigidity Theorem crucial for proving hyperbolic structure uniqueness on finite-volume hyperbolic 3-manifolds
  • Implies volume and other geometric invariants are topological invariants
• Seifert-Weber dodecahedral space hyperbolic structure proof uses fundamental polyhedron and gluing isometries
  • Constructs hyperbolic structure explicitly through polyhedron identification
• 3-torus Euclidean structure proof involves constructing it as E³ quotient by translation lattice
  • Demonstrates flat structure through explicit construction
• Non-existence proofs utilize various obstructions
  • Euler characteristic incompatibility (example: S³ geometry on manifolds with χ ≠ 0)
  • Fundamental group properties (example: non-residually finite groups for hyperbolic structures)
  • Incompatible cusp structures (example: non-Euclidean cusps for hyperbolic manifolds)

Specialized Techniques for Specific Geometries

• Seifert fibered space theory allows systematic proofs for E³, S³, S² × R, H² × R, and Nil geometries
  • Classifies Seifert fibered spaces based on base orbifold and Euler number
• Dunbar and Meyerhoff's non-compact hyperbolic 3-orbifold classification aids hyperbolic structure (non-)existence proofs
  • Provides complete list of possible non-compact hyperbolic 3-orbifolds
• Algebraic methods prove existence of Sol geometry on certain mapping tori
  • Uses properties of Anosov diffeomorphisms of tori
• Geometric group theory techniques prove existence of SL(2,R) geometry on certain circle bundles
  • Utilizes properties of surface group extensions