Closed manifold

Properties

Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups.

If ${\displaystyle M}$ is a closed connected n-manifold, the n-th homology group ${\displaystyle H_n(M;\mathbb{Z})}$ is ${\displaystyle \mathbb{Z}}$ or 0 depending on whether ${\displaystyle M}$ is orientable or not. Moreover, the torsion subgroup of the (n-1)-th homology group ${\displaystyle H_{n-1}(M;\mathbb{Z})}$ is 0 or ${\displaystyle {\mathbb{Z}}_2}$ depending on whether ${\displaystyle M}$ is orientable or not. This follows from an application of the universal coefficient theorem.

Let ${\displaystyle R}$ be a commutative ring. For ${\displaystyle R}$-orientable ${\displaystyle M}$ with fundamental class ${\displaystyle [M]\in H_n(M;R)}$, the map ${\displaystyle D:H^k(M;R)\to H_{n-k}(M;R)}$ defined by ${\displaystyle D(\alpha )=[M]\cap \alpha }$ is an isomorphism for all k. This is the Poincaré duality. In particular, every closed manifold is ${\displaystyle {\mathbb{Z}}_2}$-orientable. So there is always an isomorphism ${\displaystyle H^k(M;{\mathbb{Z}}_2)\cong H_{n-k}(M;{\mathbb{Z}}_2)}$.

Open manifolds

For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.

Abuse of language

Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical conditions), thus by this definition a manifold does not include its boundary when it is embedded in a larger space. However, this definition doesn’t cover some basic objects such as a closed disk, so authors sometimes define a manifold with boundary and abusively say manifold without reference to the boundary. But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used.

The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and it is a manifold, but not a closed manifold.

Use in physics

The notion of a "closed universe" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive Ricci curvature.