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Print/export Download as PDF; Printable version; In other projects ... For more examples see 4-manifold. Special types of manifolds. Manifolds related to spheres
In Romance languages, manifold is translated as "variety" – such spaces with a differentiable structure are literally translated as "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties". Thus for example, the French word "variété topologique" means topological manifold.
In geometry, if X is a manifold with an action of a topological group G by analytical diffeomorphisms, the notion of a (G, X)-structure on a topological space is a way to formalise it being locally isomorphic to X with its G-invariant structure; spaces with a (G, X)-structure are always manifolds and are called (G, X)-manifolds.
An example is the sphere, which can be defined as the zero set of the polynomial x 2 + y 2 + z 2 – 1, and hence is an algebraic variety. For an algebraic manifold, the ground field will be the real numbers or complex numbers; in the case of the real numbers, the manifold of real points is sometimes called a Nash manifold.
Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. [1] However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.
Example 1. If closed 2-manifolds M and N are homotopically equivalent then they are homeomorphic. Moreover, any homotopy equivalence of closed surfaces deforms to a homeomorphism. Example 2. If a closed manifold M n (n ≠ 3) is homotopy-equivalent to S n then M n is homeomorphic to S n.