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