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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.
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 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.
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 ...
Some illustrative examples of non-orientable manifolds include: (1) the Möbius strip, which is a manifold with boundary, (2) the Klein bottle, which must intersect itself in its 3-space representation, and (3) the real projective plane, which arises naturally in geometry.
Hyperbolic geometry is the most rich and least understood of the eight geometries in dimension 3 (for example, for all other geometries it is not hard to give an explicit enumeration of the finite-volume manifolds with this geometry, while this is far from being the case for hyperbolic manifolds).
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