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What is the importance of the Poincaré conjecture?
WEBThe poincare conjecture is part of a similar classification effort, but for closed 3-manifolds. Now, closed 2-manifolds have a well-understood classification in a few senses; there is a topological classification, and there is also a geometric classification. These classification results allow us to tackle many problems involving surfaces, such
Math.stackexchange.comSmooth Poincaré Conjecture - Mathematics Stack Exchange
WEBIt basically implies that if you connect sum enough non-rational spheres, your manifold cannot support a non-negatively curved metric. Also relevant is the Bott conjecture which says that a simply connected compact non-negatively curved manifold should be rationally elliptic. If the Bott conjecture is true, then. – Jason DeVito - on hiatus.
Math.stackexchange.comWhat's the best current proof of the Poincaré conjecture?
WEBDec 3, 2020 · Still, the only (essentially) self-contained treatment of Perelman's proof of the poincare conjecture is. Morgan, John; Tian, Gang, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-4328-4/hbk). xlii, 521 p
Math.stackexchange.comAre there big implications of Poincare conjecture?
WEBJul 26, 2014 at 12:13. Indeed, there is a even the s-cobordism theorem in dimension 3: It is a combination of the poincare conjecture and and old theorem of John Stallings from 1960s. However, unlike in higher dimensions, one does not prove the poincare conjecture using h-cobordism theorem but it is the other way around. – Moishe Kohan.
Math.stackexchange.comPoincare conjecture for n=2 - Mathematics Stack Exchange
WEBJan 28, 2012 · 5. Poincare for n = 2 n = 2 is contained in the classification theorem for surfaces (and that phrase should get you started, if you want to search for a proof), which says that every compact surface is homeomorphic to a sphere with some number of handles or cross-caps attached. I once heard an expert "explain" the difficulty of the n = …
Math.stackexchange.compoincare conjecture understanding - Mathematics Stack Exchange
WEBMay 9, 2018 · 1. I have knowledge of basic school math, and in colleges I have read calculus (mostly forgotten now). I need to understand poincare conjecture, and hence I need to study a lot of things. I need to know from you what I should learn step by step to understand it fully if I have basic math knowledge. Thanks.
Math.stackexchange.comHomotopy Poincaré conjecture - Mathematics Stack Exchange
WEBMay 17, 2016 · $\newcommand{\Z}{\mathbb{Z}}$ In Terence Tao's notes on page 18, concerning the Poincaré conjecture, he gave the following sketchy proof of the homotopy Poincaré conjecture.
Math.stackexchange.comsoft question - Why is four dimensions more difficult?
WEBJun 5, 2018 · $\begingroup$ The smooth poincare conjecture is not "valid for all but 4 dimensions", but rather "reduced to homotopy theory in all but dimensions $\leq 4$", and then solved for dimensions $\leq 3$. It is unknown but plausible whether every sphere of dimension larger than, say, $128$, has at least two smooth structures. $\endgroup$
Math.stackexchange.comgeometric topology - Poincaré conjectures for other shapes
WEBApr 13, 2017 · Motivation: mainly interested in shapes that answer the question: Topological shape of sphere-like equations with complex radius to see if that would lead to a Holomorphic Poincaré conjecture for such shape. That depends on what you mean by "some kinds of holes." Whatever kinds of holes that would make the conjectures true.
Math.stackexchange.comgeneral topology - Poincare conjecture 'loop' contraction issue
WEBJun 19, 2015 · According to WikiPedia's entry on Poincare's conjecture, the first image (on the right - five spheres):. For compact 2-dimensional surfaces without boundary, if every loop can be continuously tightened to a point, then the surface is topologically homeomorphic to a 2-sphere (usually just called a sphere).
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