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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.
There is an alternate definition due to Mikhail Gromov. Gromov's topology utilizes the Gromov-Hausdorff metric and is defined on pointed hyperbolic 3-manifolds. One essentially considers better and better bi-Lipschitz homeomorphisms on larger and larger balls. This results in the same notion of convergence as above as the thick part is always ...
In particular if the surgery coefficient is of the form /, then the surgered 3-manifold is still the 3-sphere. If M {\displaystyle M} is the 3-sphere, L {\displaystyle L} is the right-handed trefoil knot , and the surgery coefficient is + 1 {\displaystyle +1} , then the surgered 3-manifold is the Poincaré dodecahedral space .
The surgery structure set of a compact -dimensional manifold is a pointed set which classifies -dimensional manifolds within the homotopy type of . The basic idea is that in order to calculate S ( X ) {\displaystyle {\mathcal {S}}(X)} it is enough to understand the other terms in the sequence, which are usually easier to determine.
More precisely, Thurston's hyperbolic Dehn surgery theorem implies that a manifold with cusps is a limit of a sequence of manifolds with cusps for any <, so that the isolated points are volumes of compact manifolds, the manifolds with exactly one cusp are limits of compact manifolds, and so on. Together with results of Jørgensen the theorem ...
The geometry and topology of three-manifolds is a set of widely circulated notes for a graduate course taught at Princeton University by William Thurston from 1978 to 1980 describing his work on 3-manifolds. They were written by Thurston, assisted by students William Floyd and Steven Kerchoff. [1]
Created Date: 8/30/2012 4:52:52 PM
Let M be a small 3-manifold. Let φ be a non-trivial periodic orientation-preserving diffeomorphism of M. Then M admits a φ-invariant hyperbolic or Seifert fibered structure. This theorem is a special case of Thurston's orbifold theorem, announced without proof in 1981; it forms part of his geometrization conjecture for 3-manifolds.