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Four-manifolds often admit many unusual differentiable structures, most strikingly the uncountably infinitely many exotic differentiable structures on R 4. Similarly, differentiable 4-manifolds is the only remaining open case of the generalized Poincaré conjecture.
A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions [clarification needed]: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate ...
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.
A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closed or open manifold. For example, in the Friedmann–Lemaître–Robertson–Walker (FLRW) model, the universe is considered to be without boundaries, in which case "compact universe" could describe a universe that is a closed manifold.
A manifold need not be connected, but every manifold M is a disjoint union of connected manifolds. These are just the connected components of M, which are open sets since manifolds are locally-connected. Being locally path connected, a manifold is path-connected if and only if it is connected. It follows that the path-components are the same as ...
A contractible manifold is one that can continuously be shrunk to a point inside the manifold itself. For example, an open ball is a contractible manifold. All manifolds homeomorphic to the ball are contractible, too. One can ask whether all contractible manifolds are homeomorphic to a ball. For dimensions 1 and 2, the answer is classical and ...
If M is an m-dimensional manifold and N is an n-dimensional manifold then for an immersion f : M → N in general position the set of k-tuple points is an (n − k(n − m))-dimensional manifold. Every embedding is an immersion without multiple points (where k > 1). Note, however, that the converse is false: there are injective immersions that ...
An open surface with x-, y-, and z-contours shown.. In the part of mathematics referred to as topology, a surface is a two-dimensional manifold.Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball.