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.

Comments

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.

References

[a1]	R.D. Edwards, "The solution of the -dimensional annulus conjecture (after Frank Quinn)" Contemporary Math. , 35 (1984) pp. 211–264
[a2]	F. Quinn, "Ends of maps III: dimensions and " J. Diff. Geom. , 17 (1982) pp. 503–521
[a3]	R. Kirby, "Stable homeomorphisms and the annulus conjecture" Ann. of Math. , 89 (1969) pp. 575–582
[a4a]	E.E. Moise, "Affine structures in -manifolds I" Ann. of Math. , 54 (1951) pp. 506–533
[a4b]	E.E. Moise, "Affine structures in -manifolds II, III" Ann. of Math. , 55 (1952) pp. 172–176; 203–222
[a4c]	E.E. Moise, "Affine structures in -manifolds IV" Ann. of Math. , 56 (1952) pp. 96–114
[a5]	M. Brown, H. Gluck, "Stable structures on manifolds I-III" Ann. of Math. , 79 (1974) pp. 1–58
[a6]	T. Radó, "Über den Begriff der Riemannsche Fläche" Acta Univ. Szeged , 2 (1924–1926) pp. 101–121