Annulus theorem

In mathematics, the annulus theorem (formerly called the annulus conjecture) states roughly that the region between two well-behaved spheres is an annulus. It is closely related to the stable homeomorphism conjecture (now proved) which states that every orientation-preserving homeomorphism of Euclidean space is stable.

Statement

If S and T are topological spheres in Euclidean space, with S contained in T, then it is not true in general that the region between them is an annulus, because of the existence of wild spheres in dimension at least 3. So the annulus theorem has to be stated to exclude these examples, by adding some condition to ensure that S and T are well behaved. There are several ways to do this.

The annulus theorem states that if any homeomorphism h of Rⁿ to itself maps the unit ball B into its interior, then B − h(interior(B)) is homeomorphic to the annulus Sⁿ⁻¹×[0,1].

History of proof

The annulus theorem is trivial in dimensions 0 and 1. It was proved in dimension 2 by Radó (1924), in dimension 3 by Moise (1952), in dimension 4 by Quinn (1982), and in dimensions at least 5 by Kirby (1969).

Torus trick

Robion Kirby's torus trick is a proof method employing an immersion of a punctured torus $\mathbb {T} ^{n}-\mathbb {D} ^{n}$ into $\mathbb {R} ^{n}$, where then smooth structures can be pulled back along the immersion and be lifted to covers. The torus trick is used in Kirby's proof of the annulus theorem in dimensions $n\geq 5$. It was also employed in further investigations of topological manifolds with Laurent C. Siebenmann

Here is a list of some further applications of the torus trick that appeared in the literature:

• Proving existence and uniqueness (up to isotopy) of smooth structures on surfaces
• Proving existence and uniqueness (up to isotopy) of PL structures on 3-manifolds

The stable homeomorphism conjecture

A homeomorphism of Rⁿ is called stable if it is the composite of (a finite family of) homeomorphisms each of which is the identity on some non-empty open set.

The stable homeomorphism conjecture states that every orientation-preserving homeomorphism of Rⁿ is stable. Brown & Gluck (1964) previously showed that the stable homeomorphism conjecture is equivalent to the annulus conjecture, so it is true.

References

• Brown, Morton; Gluck, Herman (1964), "Stable structures on manifolds. II. Stable manifolds.", Annals of Mathematics, Second Series, 79 (1): 18–44, doi:10.2307/1970481, ISSN 0003-486X, JSTOR 1970482, MR 0158383
• Edwards, Robert D. (1984), "The solution of the 4-dimensional annulus conjecture (after Frank Quinn)", Four-manifold theory (Durham, N.H., 1982), Contemp. Math., vol. 35, Providence, R.I.: Amer. Math. Soc., pp. 211–264, doi:10.1090/conm/035/780581, ISBN 9780821850336, MR 0780581
• Kirby, Robion C. (1969), "Stable homeomorphisms and the annulus conjecture", Annals of Mathematics, Second Series, 89 (3): 575–582, doi:10.2307/1970652, ISSN 0003-486X, JSTOR 1970652, MR 0242165
• Moise, Edwin E. (1952), "Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung", Annals of Mathematics, Second Series, 56 (1): 96–114, doi:10.2307/1969769, ISSN 0003-486X, JSTOR 1969769, MR 0048805
• Quinn, Frank (1982), "Ends of maps. III. Dimensions 4 and 5", Journal of Differential Geometry, 17 (3): 503–521, doi:10.4310/jdg/1214437139, ISSN 0022-040X, MR 0679069
• Radó, T. (1924), "Über den Begriff der Riemannschen Fläche", Acta Univ. Szeged, 2: 101–121

Further reading

• MathOverflow discussion on the Torus trick
• Video recording of interview with Robion Kirby
• Topological Manifolds Seminar (University of Bonn, 2021)