3-manifold

A topological space locally homeomorphic to three-dimensional Euclidean space at every point. It may be compact or noncompact, and may have a boundary if that is explicitly allowed.

Topological space: A set equipped with a topology, specifying which subsets count as open. A 3-manifold is a topological space with a specific local form.
Heegaard splitting: A decomposition of a closed 3-manifold into two handlebodies joined along their boundary surface. It represents a closed 3-manifold by two simple pieces and their gluing.
3-sphere: The unit sphere in four-dimensional Euclidean space, a simply connected closed 3-manifold. It is the simplest closed 3-manifold and the subject of the Poincaré conjecture.
Geometrization conjecture: Thurston’s conjecture that every compact 3-manifold can be decomposed into pieces carrying one of eight geometries. Its proof by Perelman gives a structural framework for understanding 3-manifolds.
2-manifold: A topological space locally homeomorphic to the plane, such as a surface. Surface classification is far more complete than classification in dimension three.
Euclidean space: A space whose points have coordinates and whose geometry is determined by the Euclidean metric. Three-dimensional Euclidean space supplies the local model for every point.
Dehn surgery: A construction that removes a solid-torus neighborhood of a knot and replaces it by another solid torus. Many 3-manifolds can be built by surgery on links in the 3-sphere.
3-torus: The product of three circles, a compact 3-manifold with fundamental group isomorphic to the integers cubed. It demonstrates that a closed 3-manifold can have substantial nontrivial topology.
Poincaré conjecture: The theorem that every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. It resolves a fundamental question about recognizing the 3-sphere among 3-manifolds.
4-manifold: A topological space locally homeomorphic to four-dimensional Euclidean space. Dimension four has distinctive phenomena absent from the topology of 3-manifolds.

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