Torus theorem

This article deals with Scott's torus theorem in the 3-manifold theory. For the torus theorem about embedded tori in Euclidean space see Alexander's torus theorem .

The torus theorem is a theorem from the mathematical field of topology . It is required for the JSJ decomposition of 3-manifolds and is therefore of fundamental importance in 3-dimensional topology.

formulation

Let it be an orientable irreducible 3-manifold whose fundamental group contains a subgroup isomorphic to . Then there is either a Seifert fiber or there is an embedded incompressible torus . ${\displaystyle M}$ ${\displaystyle \mathbb {Z} \oplus \mathbb {Z} }$ ${\displaystyle M}$ ${\displaystyle T^{2}\subset M}$

Remarks

• A subgroup of the fundamental group that is too isomorphic exists if and only if there is an -injective immersion . The torus theorem can also be formulated as follows: if there is an immersed, incompressible torus, then either there is a Seifert fiber or there is an embedded incompressible torus. ${\displaystyle \mathbb {Z} \oplus \mathbb {Z} }$ ${\displaystyle \pi _{1}}$ ${\displaystyle T^{2}\to M}$ ${\displaystyle M}$ ${\displaystyle M}$
• Seifert fibers generally have numerous immersed, but not embedded, incompressible tori. These arise as follows: let be the projection image of the Seifert fiber and an embedded curve that is not null homotop . Then there is an immersed, incompressible torus which, however, generally does not have to be embedded if the fiber is singular. ${\displaystyle p\colon M\to \Sigma }$ ${\displaystyle \gamma }$ ${\displaystyle \Sigma }$ ${\displaystyle p^{-1}(\gamma )}$
• From the torus theorem it follows through counterposition : an orientable, irreducible, homotopically atoroidal 3-manifold is either a Seifert fiber or geometrically atoroidal . This formulation is important for the JSJ decomposition of 3-manifolds, it implies that the components of this decomposition are either Seifert fibers or geometrically atoroidal.

history

The torus theorem in the special case of hook manifolds was conjectured by Waldhausen in 1968 and proved by Feustel in 1976. The general case was proven by Scott in 1980 . The version originally proven by Scott stated that under the assumptions of the torus theorem, it either contains an embedded incompressible torus or a nontrivial normal cyclic subgroup . Together with the Seifert fiber space conjecture , which was proven in the 1990s, the above formulation follows. ${\displaystyle M}$ ${\displaystyle \pi _{1}M}$

Individual evidence

1. ↑ Friedhelm Waldhausen : On the determination of some bounded 3-manifolds by their fundamental groups alone. In: Proceedings of the International Symposium on Topology and Its Applications Herceg-Novi, August 25-31, 1968 Yugoslavia. = Trudy Meždunarodnogo Simpozija po Topologii i ee Primenenijach Cherceg-Novi, August 25–31, 1968, Jugoslavija. Savez društava matematičara, fizičara i astronoma Jugoslavije, Belgrad 1969, pp. 331-332.
2. ^ Charles D. Feustel: On the torus theorem and its applications. In: Transactions of the American Mathematical Society . Vol. 217, 1976, pp. 1-43, doi : 10.2307 / 1997556 .
3. ↑ Peter Scott : A new proof of the annulus and torus theorems. In: American Journal of Mathematics . Vol. 102, No. 2, 1980, pp. 241-277, doi : 10.2307 / 2374238 .