View source for 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|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 $, 
[[#References|[a6]]];  $  n= 3 $, 
;  $  n \geq  5 $, 
[[#References|[a3]]]; and, finally,  $  n= 4 $, 
[[#References|[a2]]], as an application of a special controlled [[H-cobordism| $  h $-
cobordism]] theorem in dimension  $  5 $, 
called the thin  $  h $-
cobordism theorem or Quinn's thin  $  h $-
cobordism theorem.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.D. Edwards,   "The solution of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450033.png" />-dimensional annulus conjecture (after Frank Quinn)"  ''Contemporary Math.'' , '''35'''  (1984)  pp. 211–264</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Quinn,   "Ends of maps III: dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450035.png" />"  ''J. Diff. Geom.'' , '''17'''  (1982)  pp. 503–521</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Kirby,   "Stable homeomorphisms and the annulus conjecture"  ''Ann. of Math.'' , '''89'''  (1969)  pp. 575–582</TD></TR><TR><TD valign="top">[a4a]</TD> <TD valign="top">  E.E. Moise,   "Affine structures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450036.png" />-manifolds I"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 506–533</TD></TR><TR><TD valign="top">[a4b]</TD> <TD valign="top">  E.E. Moise,   "Affine structures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450037.png" />-manifolds II, III"  ''Ann. of Math.'' , '''55'''  (1952)  pp. 172–176; 203–222</TD></TR><TR><TD valign="top">[a4c]</TD> <TD valign="top">  E.E. Moise,   "Affine structures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450038.png" />-manifolds IV"  ''Ann. of Math.'' , '''56'''  (1952)  pp. 96–114</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Brown,   H. Gluck,   "Stable structures on manifolds I-III"  ''Ann. of Math.'' , '''79'''  (1974)  pp. 1–58</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T. Radó,   "Über den Begriff der Riemannsche Fläche"  ''Acta Univ. Szeged'' , '''2'''  (1924–1926)  pp. 101–121</TD></TR></table>