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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.
In particular if the surgery coefficient is of the form /, then the surgered 3-manifold is still the 3-sphere. If is the 3-sphere, is the right-handed trefoil knot, and the surgery coefficient is +, then the surgered 3-manifold is the Poincaré dodecahedral space.
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 ...
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.
Depending on the precise definition and the category of manifolds (smooth, PL, or topological), there are various versions of structure sets. Since, by the s-cobordism theorem, certain bordisms between manifolds are isomorphic (in the respective category) to cylinders, the concept of structure set allows a classification even up to diffeomorphism.
A prism manifold is a closed 3-dimensional manifold M whose fundamental group is a central extension of a dihedral group.. The fundamental group π 1 (M) of M is a product of a cyclic group of order m with a group having presentation
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Once a small subfield of geometric topology, the theory of 3-manifolds has experienced tremendous growth in the latter half of the 20th century. The methods used tend to be quite specific to three dimensions, since different phenomena occur for 4-manifolds and higher dimensions.