Low-dimensional topology: 3- and 4-manifolds

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Requisitos previos obligatorios, hasta el fondo: 56

Aprenderás a

• Describe 3-manifolds by Heegaard splittings and Dehn surgery on links, with lens spaces as examples.
• State the geometrisation theorem and the Poincaré conjecture, and explain the role of the eight Thurston geometries.
• Contrast topological and smooth 4-manifolds through Freedman's classification and Donaldson's theorem, and identify open problems such as the smooth 4-dimensional Poincaré conjecture.

Definición

• A Heegaard splitting of a closed orientable 3-manifold M is a decomposition of M into two handlebodies glued along their common boundary surface.
• Dehn surgery on a knot removes a tubular neighbourhood of the knot and glues a solid torus back in by a different boundary homeomorphism.
• A hyperbolic 3-manifold is a 3-manifold with a complete Riemannian metric of constant sectional curvature −1.

Teorema

• Every closed orientable 3-manifold has a Heegaard splitting and arises by surgery on a link in S³.
• Perelman (2002–2003): Thurston's geometrisation conjecture holds; in particular every closed simply connected 3-manifold is homeomorphic to S³.
• Mostow rigidity: a closed hyperbolic manifold of dimension at least 3 is determined up to isometry by its fundamental group.
• ℝ⁴ admits exotic smooth structures (Freedman, Donaldson, Gompf, Taubes); whether S⁴ admits an exotic smooth structure is open.

Errores frecuentes

• Thinking the Poincaré conjecture was the last open problem in 3-manifold topology; many questions remain, though geometrisation transformed the field.
• Believing results about manifolds work the same way in every dimension; dimensions 3 and 4 behave differently from dimensions 5 and above.

Descubrimientos opcionales

• Gauge theory and the geometry of physics Donaldson used Yang–Mills instantons, and later Seiberg–Witten theory, to discover smooth 4-manifold phenomena invisible to classical topology.
• Geometric group theory The fundamental groups of closed hyperbolic 3-manifolds are word-hyperbolic groups; Agol's proof of the virtual Haken conjecture used Wise's theory of special cube complexes.
• ← Knot theory
• ← Algebraic topology
• ← Differential topology
• ← Geometric analysis
• ← Gauge theory and the geometry of physics

Itinerarios de aprendizaje

• Geometry and topology

Lecturas complementarias

• William P. Thurston, Three-Dimensional Geometry and Topology, Volume 1 (1997)
• Jeffrey R. Weeks, The Shape of Space (1985)
• Robert E. Gompf and András I. Stipsicz, 4-Manifolds and Kirby Calculus (1999)