List of geometric topology topics

This is a list of geometric topology topics.

Low-dimensional topology

Knot theory

• Knot (mathematics)
• Link (knot theory)
• Wild knots
• Examples of knots
  • Unknot
  • Trefoil knot
  • Figure-eight knot (mathematics)
  • Borromean rings
• Types of knots
  • Torus knot
  • Prime knot
  • Alternating knot
  • Hyperbolic link
• Knot invariants
  • Crossing number
  • Linking number
  • Skein relation
  • Knot polynomials
    • Alexander polynomial
    • Jones polynomial
  • Knot group
• Writhe
• Quandle
• Seifert surface
• Braids
  • Braid theory
  • Braid group
• Kirby calculus

Surfaces

• Genus (mathematics)
• Examples
  • Positive Euler characteristic
    • 2-disk
    • Sphere
    • Real projective plane
  • Zero Euler characteristic
    • Annulus
    • Möbius strip
    • Torus
    • Klein bottle
  • Negative Euler characteristic
    • The boundary of the pretzel is a genus three surface
  • Embedded/Immersed in Euclidean space
    • Cross-cap
    • Boy's surface
    • Roman surface
    • Steiner surface
    • Alexander horned sphere
    • Klein bottle
• Mapping class group
  • Dehn twist
  • Nielsen–Thurston classification

Three-manifolds

• Moise's Theorem (see also Hauptvermutung)
• Poincaré conjecture
  • Thurston elliptization conjecture
• Thurston's geometrization conjecture
  • Hyperbolic 3-manifolds
  • Spherical 3-manifolds
  • Euclidean 3-manifolds, Bieberbach Theorem, Flat manifolds, Crystallographic groups
  • Seifert fiber space
• Heegaard splitting
  • Waldhausen conjecture
  • Compression body
  • Handlebody
• Incompressible surface
  • Dehn's lemma
  • Loop theorem (aka the Disk theorem)
  • Sphere theorem
• Haken manifold
• JSJ decomposition
• Branched surface
• Lamination
• Examples
  • 3-sphere
  • Torus bundles
  • Surface bundles over the circle
  • Graph manifolds
  • Knot complements
  • Whitehead manifold
• Invariants
  • Fundamental group
  • Heegaard genus
  • tri-genus
  • Analytic torsion

Manifolds in general

• Orientable manifold
• Connected sum
• Jordan-Schönflies theorem
• Signature (topology)
• Handle decomposition
• Handlebody
• h-cobordism theorem
• s-cobordism theorem
• Manifold decomposition
• Hilbert-Smith conjecture
• Mapping class group
• Orbifolds
• Examples
  • Exotic sphere
  • Homology sphere
  • Lens space
  • I-bundle

See also

• topology glossary
• List of topology topics
• List of general topology topics
• List of algebraic topology topics
• Publications in topology

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Topology

Fields

• General (point-set)
• Algebraic
• Combinatorial
• Continuum
• Differential
• Geometric
  • low-dimensional
• Homology
  • cohomology
• Set-theoretic
• Digital

Key concepts

• Open set / Closed set
• Interior
• Continuity
• Space
  • compact
  • connected
  • Hausdorff
  • metric
  • uniform
• Homotopy
  • homotopy group
  • fundamental group
• Simplicial complex
• CW complex
• Polyhedral complex
• Manifold
• Bundle (mathematics)
• Second-countable space
• Cobordism

Metrics and properties

• Euler characteristic
• Betti number
• Winding number
• Chern number
• Orientability

Key results

• Banach fixed-point theorem
• De Rham cohomology
• Invariance of domain
• Poincaré conjecture
• Tychonoff's theorem
• Urysohn's lemma

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