Closed manifold

In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components. The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RPn is a closed n-dimensional manifold. The complex projective space CPn is a closed 2n-dimensional manifold. A line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups. If is a closed connected n-manifold, the n-th homology group is or 0 depending on whether is orientable or not. Moreover, the torsion subgroup of the (n-1)-th homology group is 0 or depending on whether is orientable or not. This follows from an application of the universal coefficient theorem. Let be a commutative ring. For -orientable with fundamental class , the map defined by is an isomorphism for all k. This is the Poincaré duality. In particular, every closed manifold is -orientable. So there is always an isomorphism . 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. 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.