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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 ...
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
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]
Manifolds need not be closed; thus a line segment without its end points is a manifold. They are never countable, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola, and the locus of points on a cubic curve y 2 = x 3 − x (a closed loop piece and an open, infinite ...
A 3-manifold is said to be small if it is closed, irreducible and does not contain any incompressible surfaces. Orbifold Theorem. 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.
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There are two usual ways to give a classification: explicitly, by an enumeration, or implicitly, in terms of invariants. For instance, for orientable surfaces, the classification of surfaces enumerates them as the connected sum of tori, and an invariant that classifies them is the genus or Euler characteristic.