3manifolds

A 3-manifold is a topological manifold of dimension three: a Hausdorff, second-countable space in which every point has a neighborhood homeomorphic to Euclidean 3-space R^3. If all points have such neighborhoods even near the boundary, the manifold is said to have boundary; otherwise it is closed.

3-manifolds may be compact or non-compact, and may be orientable or non-orientable. They can be given smooth structures, enabling the use of differential methods.

Examples include the 3-sphere S^3, Euclidean space R^3, the product S^2×S^1, lens spaces, and the solid torus. Constructions include connected sums and taking quotients by group actions.

Major results concern geometry and classification. Poincaré conjecture asserts that every closed simply connected 3-manifold is homeomorphic to S^3. Geometrization theorem, proved by Grigori Perelman (building on Thurston’s program), states that every compact 3-manifold can be decomposed along spheres and tori into pieces that each admit a homogeneous geometry, leading to a broad geometric classification.

Decomposition results further organize 3-manifolds. Prime decomposition expresses a compact orientable 3-manifold as a connected sum of prime manifolds. JSJ decomposition cuts along incompressible tori into pieces that are either atoroidal or Seifert-fibered, revealing the geometric structure of each piece.

Invariants and applications play a central role. The fundamental group and homology provide algebraic data about a manifold, while a large class of hyperbolic 3-manifolds carry complete metrics of constant negative curvature. Many 3-manifolds arise as knot or link complements in the 3-sphere, making 3-manifold theory central to low-dimensional topology and knot theory.