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Two-dimensional annulus

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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 , where is the circle and the interval. An -dimensional annulus is a space homeomorphic to . The -dimensional annulus conjecture states that for any homeomorphism such that , the interior of , the closed difference

is homeomorphic to the annulus . Here, .

The stable homeomorphism conjecture asserts that any orientation-preserving homeomorphism can be written as a finite product, , where each is the identity on some open subset of .

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

The stable homeomorphism conjecture (and hence the annulus conjecture) has finally been established for all : , classical; , [a6]; , ; , [a3]; and, finally, , [a2], as an application of a special controlled -cobordism theorem in dimension , called the thin -cobordism theorem or Quinn's thin -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
How to Cite This Entry:
Two-dimensional annulus. A.V. Chernavskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Two-dimensional_annulus&oldid=17821
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098