Seifert fiber space conjecture

In mathematics , the Seifert fiber space conjecture is a central theorem of 3-dimensional topology proven by Casson - Jungreis and Gabai and part of the geometry of 3-manifolds .

sentence

Let be an irreducible , orientable , compact 3-manifold . If the fundamental group has a normal divisor that is an infinite cyclic group , then it is a Seifertian fiber space . ${\displaystyle M}$ ${\displaystyle \pi _{1}M}$ ${\displaystyle M}$

history

Seifert fibers were defined and classified by Seifert in the early 1930s . If a Seifert fiber is irreducible and has an infinite fundamental group, then the homotopy class of the fiber creates an infinite cyclic group in the center of the fundamental group, which is in particular a normal divisor.

Burde and Zieschang proved that a knot group only has an infinitely cyclic normal divisor if the knot is a torus knot . Since torus knots are the only knots whose complement is a Seifertian fiber space, this proves from a later point of view the conjecture for knot complements.

Waldhausen proved the conjecture for hook manifolds under the assumption that the cyclic normal divider belongs to the center of . The general proof for hook manifolds was then given by Jaco and Shalen after preliminary work by Gordon and Heil . ${\displaystyle \pi _{1}M}$

Scott proved that a closed, irreducible, orientable 3-manifold, whose fundamental group is infinite and isomorphic to the fundamental group of a Seifert fiber, must itself be a Seifert fiber. In doing so, he reduced the Seifert fiber space conjecture to a group-theoretical problem: to prove that the quotient of after the infinite cyclic group is either a Fuchs group or a 2-dimensional Euclidean crystallographic group . ${\displaystyle \pi _{1}M}$

Mess proved that the superposition associated with the infinitely cyclic normal divider is homeomorphic to . In particular, the quotient from after the infinite cyclic group is either a convergence group on the circle or a 2-dimensional Euclidean crystallographic group. In doing so, he reduced the Seifert fiber space conjecture to the question of whether every convergence group on the circle is a Fuchsian group (apart from conjugation with homeomorphisms of the circle). ${\displaystyle S^{1}\times \mathbb {R} ^{2}}$ ${\displaystyle \pi _{1}M}$

Tukia proved this conjecture about convergence groups with the exception of groups that have a torsional element of order . The remaining case was solved by Andrew Casson and Douglas Jungreis, while David Gabai gave independent evidence using entirely different methods. ${\displaystyle \geq 3}$

literature

• Gerhard Burde, Heiner Zieschang : A marking of the torus nodes. In: Mathematical Annals . Vol. 167, No. 2, 1966, pp. 169-176, ( doi : 10.1007 / BF01362170 ).
• Friedhelm Waldhausen : Groups with a center and 3-dimensional manifolds. In: Topology. Vol. 6, No. 4, 1967, pp. 505-517, doi : 10.1016 / 0040-9383 (67) 90008-0 .
• Cameron McA. Gordon , Wolfgang Heil: Cyclic normal subgroups of fundamental groups of 3-manifolds. In: Topology. Vol. 14, No. 4, 1975, pp. 305-309, doi : 10.1016 / 0040-9383 (75) 90014-2 .
• William H. Jaco , Peter B. Shalen : Seifert Fibered Spaces in 3-Manifolds (= Memoirs of the American Mathematical Society. 220). American Mathematical Society, Providence RJ 1979, ISBN 0-8218-2220-9 .
• Peter Scott : There are no fake Seifert fiber spaces with infinite . ${\displaystyle \pi _{1}}$In: Annals of Mathematics . Series 2, Vol. 117, No. 1, 1983, pp. 35-70, doi : 10.2307 / 2006970 .
• Geoffrey Mess : The Seifert conjecture and groups which are quasi-isometric to planes. Preprint (1988).
• Pekka Tukia : Homeomorphic conjugates of Fuchsian groups. In: Journal for pure and applied mathematics . Vol. 391, pp. 1-54, 1988, doi : 10.1515 / crll.1988.391.1 .
• David Gabai : Convergence groups are Fuchsian groups. In: Annals of Mathematics. Series 2, Vol. 136, No. 3, 1992, pp. 447-510, doi : 10.2307 / 2946597 .
• Andrew Casson , Douglas Jungreis: Convergence groups and Seifert fibered 3-manifolds. In: Inventiones Mathematicae . Vol. 118, 1994, pp. 441-456, doi : 10.1007 / BF01231540 .

Web links

• Jean-Philippe Préaux: A Survey on Seifert Fiber Space Theorem. In: International Scholarly Research Notices. 2014, doi : 10.1155 / 2014/694106