# Two-dimensional annulus

in topology

A topological image of the closed part of the plane comprised between two non-identical concentric circles. A two-dimensional annulus is an orientable two-dimensional manifold of genus zero with two boundary components.

Thus, a $2$- dimensional annulus is homeomorphic to $S ^ {1} \times I$, where $S ^ {1}$ is the circle and $I$ the interval. An $n$- dimensional annulus is a space homeomorphic to $S ^ {n-} 1 \times I$. The $n$- dimensional annulus conjecture states that for any homeomorphism $h: \mathbf R ^ {n} \rightarrow \mathbf R ^ {n}$ such that $h( B ^ {n} ) \subset \mathop{\rm Int} ( B ^ {n} )$, the interior of $B ^ {n}$, the closed difference

$$B ^ {n} \setminus h( \mathop{\rm Int} ( B ^ {n} ))$$

is homeomorphic to the annulus $S ^ {n-} 1 \times I$. Here, $B ^ {n} = \{ {x \in \mathbf R ^ {n} } : {\| x \| \leq 1 } \}$.

The stable homeomorphism conjecture asserts that any orientation-preserving homeomorphism $h: \mathbf R ^ {n} \rightarrow \mathbf R ^ {n}$ can be written as a finite product, $h = h _ {1} \dots h _ {m}$, where each $h _ {i}$ is the identity on some open subset of $\mathbf R ^ {n}$.

The stable homeomorphism conjecture for dimension $n$ implies the annulus conjecture for dimension $n$.

The stable homeomorphism conjecture (and hence the annulus conjecture) has finally been established for all $n$: $n= 1$, classical; $n= 2$, [a6]; $n= 3$,

$n \geq 5$,

[a3]; and, finally, $n= 4$, [a2], as an application of a special controlled $h$- cobordism theorem in dimension $5$, called the thin $h$- cobordism theorem or Quinn's thin $h$- cobordism theorem.

How to Cite This Entry:
Two-dimensional annulus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-dimensional_annulus&oldid=49050
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article