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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.
Infinite dimensional manifolds The definition of a manifold can be generalized by dropping the requirement of finite dimensionality. Thus an infinite dimensional manifold is a topological space locally homeomorphic to a topological vector space over the reals.
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 topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: drilling then filling.
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
Definition. An open book decomposition of a 3-dimensional manifold M is a pair (B, π) where . B is an oriented link in M, called the binding of the open book;; π: M \ B → S 1 is a fibration of the complement of B such that for each θ ∈ S 1, π −1 (θ) is the interior of a compact surface Σ ⊂ M whose boundary is B.
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More precisely, Thurston's hyperbolic Dehn surgery theorem implies that a manifold with cusps is a limit of a sequence of manifolds with cusps for any <, so that the isolated points are volumes of compact manifolds, the manifolds with exactly one cusp are limits of compact manifolds, and so on. Together with results of Jørgensen the theorem ...