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
, 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 ![]() |
[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