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The Alexander horned sphere is a pathological object in topology discovered by J. W. Alexander (1924).
In mathematics, an n-dimensional differential structure on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If M is already a topological manifold, it is required that the new topology be identical to the existing one.
R. H. Bing was an American mathematician who worked mainly in the areas of geometric topology and continuum theory. His father was named Rupert Henry, but Bing's mother thought that "Rupert Henry" was too British for Texas. She compromised by abbreviating it to R. H. Consequently, R. H. does not stand for a first or middle name.
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
In differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one.
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique.
In mathematics, topology generalizes the notion of triangulation in a natural way as follows:
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation.
The Hauptvermutung of geometric topology is the question of whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the same combinatorial pattern.
In mathematics, more specifically in geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a PL-structure.
In mathematics, an exotic is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds. There is a continuum of non-diffeomorphic differentiable structures of as was shown first by Clifford Taubes.
Robion Cromwell Kirby is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology. Together with Laurent C. Siebenmann he invented the Kirby–Siebenmann invariant for classifying the piecewise linear structures on a topological manifold. He also proved the fundamental result on the Kirby calculus, a method for describing 3-manifolds and smooth 4-manifolds by surgery on framed links. Along with his significant mathematical contributions, he is an influential figure in the field, with over 50 doctoral students and his famous problem list.
In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere is a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is
In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.
John Robert Stallings Jr. was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings was a Professor Emeritus in the Department of Mathematics at the University of California at Berkeley where he had been a faculty member since 1967. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds. Stallings' most important contributions include a proof, in a 1960 paper, of the Poincaré Conjecture in dimensions greater than six and a proof, in a 1971 paper, of the Stallings theorem about ends of groups.
In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem.
In geometric topology, the double suspension theorem of James W. Cannon and Robert D. Edwards states that the double suspension S2X of a homology sphere X is a topological sphere.
References
- Moise, Edwin E. (1952), "Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung", Annals of Mathematics , Second Series, 56: 96–114, doi:10.2307/1969769, ISSN 0003-486X, JSTOR 1969769, MR 0048805
- Moise, Edwin E. (1977), Geometric topology in dimensions 2 and 3, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90220-3, MR 0488059
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