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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.
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 ...
Because the exterior derivative d has the property that d 2 = 0, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold. The k -th de Rham cohomology (group) is the vector space of closed k -forms modulo the exact k -forms; as noted in the previous section, the Poincaré lemma states that these vector spaces ...
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 ...
Similarly, every smooth -dimensional manifold can be immersed in the -dimensional sphere (this removes the > constraint). The weak version, for 2 m + 1 {\displaystyle 2m+1} , is due to transversality ( general position , dimension counting ): two m -dimensional manifolds in R 2 m {\displaystyle \mathbf {R} ^{2m}} intersect generically in a 0 ...
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.
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 mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnor [1] pg 14 was trying to understand the structure of ()-connected manifolds of dimension (since -connected -manifolds are homeomorphic to spheres, this is the first non-trivial case after) and found an example of a space which is homotopy equivalent to a sphere, but was not explicitly diffeomorphic.