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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 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 .
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 theory of 3-manifolds, a compression body is a kind of generalized handlebody. A compression body is either a handlebody or the result of the following construction: Let be a compact, closed surface (not necessarily connected). Attach 1-handles to [,] along {}.
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 ...
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 ...
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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