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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.
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, a spherical 3-manifold M is a 3-manifold of the form = / where is a finite subgroup of O(4) acting freely by rotations on the 3-sphere. All such manifolds are prime, orientable, and closed. Spherical 3-manifolds are sometimes called elliptic 3-manifolds.
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.
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]
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 ...
In the 3-manifold case, a lens space can be visualized as the result of gluing two solid tori together by a homeomorphism of their boundaries. Often the 3-sphere and S 2 × S 1 {\displaystyle S^{2}\times S^{1}} , both of which can be obtained as above, are not counted as they are considered trivial special cases.
Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds. A 3-manifold is irreducible if and only if it is prime, except for two cases: the product and the non-orientable fiber bundle of the 2-sphere over the circle are both prime but not ...
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