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 -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.
|[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|
Two-dimensional annulus. A.V. Chernavskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Two-dimensional_annulus&oldid=17821