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Some authors use the term Alexander trick for the statement that every homeomorphism of can be extended to a homeomorphism of the entire ball .. However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true piecewise-linearly, but not smoothly.
Any closed topological n-ball is homeomorphic to the closed n-cube [0, 1] n. An n-ball is homeomorphic to an m-ball if and only if n = m. The homeomorphisms between an open n-ball B and R n can be classified in two classes, that can be identified with the two possible topological orientations of B. A topological n-ball need not be smooth; if it ...
As trivial examples, note that attaching a 0-handle is just taking a disjoint union with a ball, and that attaching an n-handle to (,) is gluing in a ball along any sphere component of . Morse theory was used by Thom and Milnor to prove that every manifold (with or without boundary) is a handlebody, meaning that it has an expression as a union ...
In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), [2] [3] also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.
The full classification of n-manifolds for n greater than three is known to be impossible; it is at least as hard as the word problem in group theory, which is known to be algorithmically undecidable. [10] In fact, there is no algorithm for deciding whether a given manifold is simply connected. There is, however, a classification of simply ...
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane.
We build the pushforward homomorphism as follows (for singular or simplicial homology): . First, the map : induces a homomorphism between the singular or simplicial chain complex and () defined by composing each singular n-simplex: with to obtain a singular n-simplex of , # =:, and extending this linearly via # = # ().