Ads
related to: 3 manifolds in math problems with solutions
Search results
Results from the WOW.Com Content Network
The prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds. A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension.
Perelman, Grisha (March 10, 2003). "Ricci flow with surgery on three-manifolds". arXiv: math.DG/0303109. Perelman, Grisha (July 17, 2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv: math.DG/0307245. Szpiro, George (2008). Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's ...
If is a prime 3-manifold then either it is or the non-orientable bundle over , or it is irreducible, which means that any embedded 2-sphere bounds a ball. So the theorem can be restated to say that there is a unique connected sum decomposition into irreducible 3-manifolds and fiber bundles of S 2 {\displaystyle S^{2}} over S 1 . {\displaystyle ...
In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold. The tameness theorem was conjectured by Marden (1974). It was proved by Agol (2004) and, independently, by Danny Calegari and ...
In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem. A simple and useful version of the loop theorem states that if for some 3-dimensional manifold M with boundary ∂M there ...
Manifolds need not be closed; thus a line segment without its end points is a manifold. They are never countable, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola, and the locus of points on a cubic curve y 2 = x 3 − x (a closed loop piece and an open, infinite ...
That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold. After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds. The conjecture is usually attributed to Friedhelm Waldhausen in a paper from 1968, [1] although he did not formally ...
Familiar examples of two-dimensional manifolds include the sphere, torus, and Klein bottle; this book concentrates on three-dimensional manifolds, and on two-dimensional surfaces within them. A particular focus is a Heegaard splitting, a two-dimensional surface that partitions a 3-manifold into two handlebodies. It aims to present the main ...
Ads
related to: 3 manifolds in math problems with solutions