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dbr:Prime_decomposition_of_3-manifolds
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Prime manifold 3-varietà irriducibile
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In geometria, e più precisamente nella topologia della dimensione bassa, una 3-varietà irriducibile è una 3-varietà in cui ogni sfera borda una palla. Una 3-varietà che contiene una sfera non bordante una palla è invece detta riducibile: questa può essere effettivamente "ridotta" a una varietà più semplice tramite l'operazione inversa della somma connessa. 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 irreducible n-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.
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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 irreducible n-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. In geometria, e più precisamente nella topologia della dimensione bassa, una 3-varietà irriducibile è una 3-varietà in cui ogni sfera borda una palla. Una 3-varietà che contiene una sfera non bordante una palla è invece detta riducibile: questa può essere effettivamente "ridotta" a una varietà più semplice tramite l'operazione inversa della somma connessa. Una 3-varietà è prima se non è ottenuta come somma connessa non banale di due varietà. I concetti di irriducibile e prima sono equivalenti per tutte le 3-varietà, con due sole eccezioni. L'ipotesi di irriducibilità è però più facile da esprimere e da gestire in molti casi, ed è quindi quella usata più spesso.
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