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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 ...
This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics . For categorical listings see Category:Manifolds and its subcategories.
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 ...
Later the Geometry Center at the University of Minnesota sold a loosely bound copy of the notes. In 2002, Sheila Newbery typed the notes in TeX and made a PDF file of the notes available, which can be downloaded from MSRI using the links below. The book (Thurston 1997) is an expanded version of the first three chapters of the notes. In 2022 the ...
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.
A Seifert manifold is a closed 3-manifold together with a decomposition into a disjoint union of circles (called fibers) such that each fiber has a tubular neighborhood that forms a standard fibered torus.
An atlas for a topological space is an indexed family {(,):} of charts on which covers (that is, =).If for some fixed n, the image of each chart is an open subset of n-dimensional Euclidean space, then is said to be an n-dimensional manifold.