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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.
The key part of the theorem is a construction of the de Rham homomorphism. [1] Let M be a manifold. Then there is a map : () from the space of differential p-forms to the space of smooth singular p-cochains given by
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 ...
In particular it is possible to use calculus on a differentiable manifold. Each point of an n-dimensional differentiable manifold has a tangent space. This is an n-dimensional Euclidean space consisting of the tangent vectors of the curves through the point. Two important classes of differentiable manifolds are smooth and analytic manifolds ...
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 ...
Precisely, it states that every closed p-form on an open ball in R n is exact for p with 1 ≤ p ≤ n. [1] The lemma was introduced by Henri Poincaré in 1886. [2] [3] Especially in calculus, the Poincaré lemma also says that every closed 1-form on a simply connected open subset in is exact.
The cases n = 1 and 2 have long been known by the classification of manifolds in those dimensions. For a PL or smooth homotopy n-sphere, in 1960 Stephen Smale proved for n ≥ 7 {\displaystyle n\geq 7} that it was homeomorphic to the n -sphere and subsequently extended his proof to n ≥ 5 {\displaystyle n\geq 5} ; [ 3 ] he received a Fields ...