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 - 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 ![]() |
[a2] | F. Quinn, "Ends of maps III: dimensions ![]() ![]() |
[a3] | R. Kirby, "Stable homeomorphisms and the annulus conjecture" Ann. of Math. , 89 (1969) pp. 575–582 |
[a4a] | E.E. Moise, "Affine structures in ![]() |
[a4b] | E.E. Moise, "Affine structures in ![]() |
[a4c] | E.E. Moise, "Affine structures in ![]() |
[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 |
Two-dimensional annulus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-dimensional_annulus&oldid=49050