Alexander Torus Theorem

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

In the mathematical branch of topology , Alexander's torus theorem is a theorem about knotted tori in the 3-dimensional sphere . ${\displaystyle S^{3}}$

Alexander theorem

The torus set of Alexander states: Every differentiable embedded torus

${\displaystyle T^{2}\subset S^{3}}$

edged a full torus embedded in the . ${\displaystyle S^{3}}$

Historical note: Alexander originally proved this theorem not for differentiable, but for PL embeddings.

Generalizations

Alexander's theorem also applies analogously in higher dimensions:

Be

${\displaystyle f\colon S^{p}\times S^{q}\to S^{p+q+1}}$

a differentiable embedding with . If either and , or and , then the closure is one of the two components of ${\displaystyle p,q\geq 1}$ ${\displaystyle 1\leq p\leq q}$ ${\displaystyle p+q\not =3}$ ${\displaystyle p=1}$ ${\displaystyle q=2}$

${\displaystyle S^{p+q+1}\setminus f(S^{p}\times S^{q})}$

diffeomeorph too . ${\displaystyle D^{p+1}\times S^{q}}$

In contrast, there are differentiable embedded 3-tori that do not border a submanifold that is too homeomorphic. ${\displaystyle T^{3}\subset S^{4}}$ ${\displaystyle D^{2}\times T^{2}}$

Individual evidence

1. ↑ James W. Alexander : On the subdivision of a 3-space by a polyhedron. In: Proceedings of the National Academy of Sciences of the United States of America . Vol. 10, No. 1, 1924, pp. 6-8, JSTOR 84201 .
2. ^ Antoni Kosiński: On Alexander's theorem and knotted spheres. In: Marion K. Fort (Ed.): Topology of 3-manifolds and related topics. Proceedings of the University of Georgia Institute, August 14-09, 1961. Prentice Hall, Englewood Cliffs NJ 1962, pp. 55-57.
3. ^ Charles TC Wall : Unknotting tori in codimension one and spheres in codimension two. In: Mathematical Proceedings of the Cambridge Philosophical Society . Vol. 61, No. 3, 1965, pp. 659-664, doi : 10.1017 / S0305004100039001 .
4. ↑ Richard Z. Goldstein: piecewise linear unknotting of S ^p × S ^q in S ^{p + q + 1} . In: Michigan Mathematical Journal Vol. 14, No. 4, 1967, pp. 405-415, doi : 10.1307 / mmj / 1028999841 .
5. ^ J. Hyam Rubinstein : Dehn's lemma and handle decompositions of some 4-manifolds. In: Pacific Journal of Mathematics . Vol. 86, No. 2, 1980, pp. 565-569, doi : 10.2140 / pjm.1980.86.565 .
6. ↑ Osamu Saeki, Laércio Aparecido Lucas, Manzoli Neto Oziride: A generalization of Alexander's torus theorem to higher dimensions and an unknotting theorem for S ^p × S ^q embedded in S ^{p + q + 2} . In: Kobe Journal of Mathematics. Vol. 13, No. 2, 1996, pp. 145-165.
7. ↑ Atsuko Katanaga, Osamu Saeki: Embeddings of quaternion space in S ⁴ . In: Journal of the Australian Mathematical Society. Series A: Pure Mathematics and Statistics. Vol. 65, No. 3, 1998, pp. 313-325, doi : 10.1017 / S1446788700035904 .
8. ↑ Laércio Aparecido Lucas, Osamu Saeki: Codimension one embeddings of product of three spheres. In: Topology and its Applications. Vol. 146/147, 2005, pp. 409-419, doi : 10.1016 / j.topol.2003.06.005 .