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2. 3-manifold - Wikipedia

   en.wikipedia.org/wiki/3-manifold

   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.

3. Introduction to 3-Manifolds - Wikipedia

   en.wikipedia.org/wiki/Introduction_to_3-Manifolds

   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 ...

4. Prime decomposition of 3-manifolds - Wikipedia

   en.wikipedia.org/wiki/Prime_decomposition_of_3...

   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 ...

5. Dehn surgery - Wikipedia

   en.wikipedia.org/wiki/Dehn_surgery

   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 .

6. The geometry and topology of three-manifolds - Wikipedia

   en.wikipedia.org/wiki/The_geometry_and_topology...

   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]

7. Atoroidal - Wikipedia

   en.wikipedia.org/wiki/Atoroidal

   In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus.There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non-boundary parallel, incompressible torus, or it may be defined algebraically, as a subgroup of its fundamental group that is not conjugate to a peripheral subgroup (i.e., the image of the ...