JSJ decomposition
The Jaco-Shalen-Johannson decomposition , abbreviated JSJ decomposition , named after William Jaco , Peter Shalen and Klaus Johannson , is a statement from the topology of the 3-manifolds .
statement
It says that every irreducible 3-dimensional manifold has a unique Seifert-fibred submanifold with an atoroidal complement (except for isotopic ) . This is also referred to as the characteristic submanifold .
proof
The proof was carried out in 1979 by William Jaco and Peter Shalen and, independently of them, by Klaus Johannson .
Consequences
The JSJ decomposition is an important prerequisite for the geometrization of 3-manifolds . Every Seifert-fibered manifold can be geometrized, and the Thurston conjecture proved by Grigori Perelman says that every atoroidal irreducible 3-manifold has a hyperbolic metric.
Web links
- Neumann, Swarup: Canonical Decompositions of 3-Manifolds.
swell
- ^ Jaco, William H .; Shalen, Peter B. Seifert fibered spaces in 3-manifolds. Mem. Amer. Math. Soc. 21 (1979), no.220
- ^ Johannson, Klaus, Homotopy equivalences of 3-manifolds with boundaries. Lecture Notes in Mathematics, 761. Springer, Berlin, 1979. ISBN 3-540-09714-7