Annulus theorem
Encyclopedia
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.
, 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].
The stable homeomorphism conjecture states that every orientation-preserving homeomorphism of Rn is stable. previously showed that the stable homeomorphism conjecture is equivalent to the annulus conjecture, so it is true.
Annulus (mathematics)
In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...
. 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 annulusAnnulus (mathematics)
In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...
, 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 , in dimension 3 by , in dimension 4 by , and in dimensions at least 5 by .The stable homeomorphism conjecture
A homeomorphism of Rn is called stable if it is a product 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. previously showed that the stable homeomorphism conjecture is equivalent to the annulus conjecture, so it is true.