Closed manifold
A closed manifold is a space that is locally like ordinary space (a manifold), has no boundary, and is compact (finite in extent).
Examples and non-examples
- 1-dimensional: a circle.
- 2-dimensional: the sphere, the torus, and the Klein bottle.
- Other closed examples: real projective space RP^n, and complex projective space CP^n (as a real manifold of dimension 2n).
- Not closed: a line (not compact). A closed disk is compact but has a boundary, so it is not a closed manifold.
Key facts about closed manifolds
- They have finite homology groups and are nice topologically (they are Euclidean neighborhood retracts).
- For a connected closed n-manifold M:
- The n-th homology H_n(M; Z) is Z if M is orientable and 0 if it is not.
- The torsion in the (n−1)-th homology H_{n−1}(M; Z) is 0 if M is orientable and Z_2 if M is nonorientable.
- Poincaré duality: if M is orientable (over a ring R) with a fundamental class [M], the cap product gives an isomorphism D: H^k(M; R) → H_{n−k}(M; R) for all k. In particular, with Z_2 coefficients, H^k(M; Z_2) ≅ H_{n−k}(M; Z_2). Every closed manifold is Z_2-orientable, so this mod 2 duality always holds.
Notes on terminology
- Open versus boundary: for connected manifolds, “open” means boundaryless and non-compact; for a disconnected manifold, “open” is a stronger condition. Many textbooks define a manifold as locally like Euclidean space and without boundary; to include boundary objects, they speak of manifolds with boundary.
- Compact (closed) vs closed as a set: a closed manifold is not the same as a closed set in the plane. A line is a closed set but not a closed manifold; a closed disk is compact but has a boundary, so it’s not a closed manifold.
- When people say “closed universe,” they usually mean a closed manifold in a geometry context, often related to positive curvature, but the phrase can have different uses in physics.
This page was last edited on 2 February 2026, at 08:06 (CET).