In topology, a branch of mathematics, a prime manifold is an n-manifold that cannot be expressed as a non-trivial connected sum of two n-manifolds. Non-trivial means that neither of the two is an n-sphere.
A similar notion is that of an irreduciblen-manifold, which is one in which any embedded (n − 1)-sphere bounds an embedded n-ball. 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.
The notions of irreducibility in algebra and manifold theory are related. An irreducible manifold is prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over the circle S1 and the twisted 2-sphere bundle over S1.
According to a theorem of Hellmuth Kneser and John Milnor, every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds.
branch of mathematics, a primemanifold is an n-manifold that cannot be expressed as a non-trivial connected sum of two n-manifolds. Non-trivial means that...
In topology, a branch of mathematics, a manifold M may be decomposed or split by writing M as a combination of smaller pieces. When doing so, one must...
connected sum of prime knots. The prime decomposition of 3-manifolds is another example of this type. Beyond mathematics and computing, prime numbers have...
geometrization conjecture. A 3-manifold is called closed if it is compact and has no boundary. Every closed 3-manifold has a prime decomposition: this means...
John Streeter Manifold AM (21 April 1915 – 19 April 1985) was an Australian poet and critic. He was born in Melbourne, into a well known Camperdown family...
topological manifold can be calculated. A connected topological manifold is locally homeomorphic to Euclidean n-space, in which the number n is the manifold's dimension...
mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a...
manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has...
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1 {\displaystyle n\geq 1} ...
of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction...
smooth manifold, R is the ring of smooth real functions on M, and x is a point in M, then the set of all smooth functions f with f (x) = 0 forms a prime ideal...
the Fubini–Study metric, such a manifold is always a Kähler manifold. By Chow's theorem, a projective complex manifold is also a smooth projective algebraic...
infinitely differentiable; used in calculus and topology Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions Smooth...
Algebraic number theory is also used in the study of arithmetic hyperbolic 3-manifolds. Class field theory Kummer theory Locally compact field Tamagawa number...
known as Cartan–Hadamard manifolds? Chern's conjecture (affine geometry) that the Euler characteristic of a compact affine manifold vanishes. Chern's conjecture...
M as a Riemannian manifold can be recovered from this data. This suggests that one might define a noncommutative Riemannian manifold as a spectral triple...
f ′ ( x ) , f ′ ′ ( x ) , … , f ( k ) ( x ) {\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)} exist and are continuous over the domain...
field corresponds to a closed, orientable 3-manifold Ideals in the ring of integers correspond to links, and prime ideals correspond to knots. The field Q...
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