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Alexander's Trick
Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander. Statement Two homeomorphisms of the ''n''-dimensional ball D^n which agree on the boundary sphere S^ are isotopic. More generally, two homeomorphisms of ''D''''n'' that are isotopic on the boundary are isotopic. Proof Base case: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary. If f\colon D^n \to D^n satisfies f(x) = x \text x \in S^, then an isotopy connecting ''f'' to the identity is given by : J(x,t) = \begin tf(x/t), & \text 0 \leq \, x\, 0 the transformation J_t replicates f at a different scale, on the disk of radius t, thus as t\rightarrow 0 it is reasonable to expect that J_t merges to the identity. The subtlety is that at t=0, f "disappears": the germ at the origin "jumps" from an infinitely stretched version of f to the identity. Each of the steps in the homotopy could be smoothed (smoot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of ''simple'' homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently. Differences between low-dimensional and high-dimensional topology Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topologica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Manifold
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. An isomorphism of PL manifolds is called a PL homeomorphism. Relation to other categories of manifolds PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory. Smooth manifolds Smooth manifolds have canonical PL structures — they are uniquely ''triangulizable,'' by Whitehead's theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The National Academy Of Sciences Of The United States Of America
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sciences, published since 1915, and publishes original research, scientific reviews, commentaries, and letters. According to ''Journal Citation Reports'', the journal has a 2021 impact factor of 12.779. ''PNAS'' is the second most cited scientific journal, with more than 1.9 million cumulative citations from 2008 to 2018. In the mass media, ''PNAS'' has been described variously as "prestigious", "sedate", "renowned" and "high impact". ''PNAS'' is a delayed open access journal, with an embargo period of six months that can be bypassed for an author fee ( hybrid open access). Since September 2017, open access articles are published under a Creative Commons license. Since January 2019, ''PNAS'' has been online-only, although print issues are ava ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clutching Construction
In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres. Definition Consider the sphere S^n as the union of the upper and lower hemispheres D^n_+ and D^n_- along their intersection, the equator, an S^. Given trivialized fiber bundles with fiber F and structure group G over the two hemispheres, then given a map f\colon S^ \to G (called the ''clutching map''), glue the two trivial bundles together via ''f''. Formally, it is the coequalizer of the inclusions S^ \times F \to D^n_+ \times F \coprod D^n_- \times F via (x,v) \mapsto (x,v) \in D^n_+ \times F and (x,v) \mapsto (x,f(x)(v)) \in D^n_- \times F: glue the two bundles together on the boundary, with a twist. Thus we have a map \pi_ G \to \text_F(S^n): clutching information on the equator yields a fiber bundle on the total space. In the case of vector bundles, this yields \pi_ O(k) \to \text_k(S^n), and indeed this map is an isomorphism (under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
