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A differentiable manifold (of class C k) consists of a pair (M, O M) where M is a second countable Hausdorff space, and O M is a sheaf of local R-algebras defined on M, such that the locally ringed space (M, O M) is locally isomorphic to (R n, O). In this way, differentiable manifolds can be thought of as schemes modeled on R n.
Call an open cover of a manifold a "de Rham cover", if all elements of the cover are de Rham, as well as all of their finite intersections. One shows that convex sets in R n {\displaystyle \mathbb {R} ^{n}} are de Rham, basically by the homotopy invariance of both cohomologies in question.
In the language of cohomology, the Poincaré lemma says that the k-th de Rham cohomology group of a contractible open subset of a manifold M (e.g., =) vanishes for . In particular, it implies that the de Rham complex yields a resolution of the constant sheaf R M {\displaystyle \mathbb {R} _{M}} on M .
For compact manifolds, results depend on the complexity of the manifold as measured by the second Betti number b 2. For large Betti numbers b 2 > 18 in a simply connected 4-manifold, one can use a surgery along a knot or link to produce a new differential structure. With the help of this procedure one can produce countably infinite many ...
Recall that a topological manifold is a topological space which is locally homeomorphic to . Differentiable manifolds generalize the notion of smoothness on in the following sense: a differentiable manifold is a topological manifold with a differentiable atlas, i.e. a collection of maps from open subsets of to the manifold which are used to "pull back" the differential structure from to the ...
Lens spaces are a class of differentiable manifolds that are quotients of odd-dimensional spheres. Lie groups are a class of differentiable manifolds equipped with a compatible group structure. The E8 manifold is a topological manifold which cannot be given a differentiable structure.
More generally the notion of isomorphism differs between the categories Top, PL, and Diff. It is the same in dimension 3 and below. In dimension 4, PL and Diff agree, but Top differs. In dimensions above 6 they all differ. In dimensions 5 and 6 every PL manifold admits an infinitely differentiable structure that is so-called Whitehead ...
The objects of Man • p are pairs (,), where is a manifold along with a basepoint , and its morphisms are basepoint-preserving p-times continuously differentiable maps: e.g. : (,) (,), such that () =. [1] The category of pointed manifolds is an example of a comma category - Man • p is exactly ({}), where {} represents an arbitrary singleton ...