View source for 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|two-dimensional manifold]] of genus zero with two boundary components.



====Comments====
Thus, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t0945001.png" />-dimensional annulus is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t0945002.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t0945003.png" /> is the circle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t0945004.png" /> the interval. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t0945006.png" />-dimensional annulus is a space homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t0945007.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t0945008.png" />-dimensional annulus conjecture states that for any homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t0945009.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450010.png" />, the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450011.png" />, the closed difference

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450012.png" /></td> </tr></table>

is homeomorphic to the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450013.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450014.png" />.

The stable homeomorphism conjecture asserts that any orientation-preserving homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450015.png" /> can be written as a finite product, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450016.png" />, where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450017.png" /> is the identity on some open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450018.png" />.

The stable homeomorphism conjecture for dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450019.png" /> implies the annulus conjecture for dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450020.png" />.

The stable homeomorphism conjecture (and hence the annulus conjecture) has finally been established for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450021.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450022.png" />, classical; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450023.png" />, [[#References|[a6]]]; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450024.png" />, ; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450025.png" />, [[#References|[a3]]]; and, finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450026.png" />, [[#References|[a2]]], as an application of a special controlled [[H-cobordism|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450027.png" />-cobordism]] theorem in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450028.png" />, called the thin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450030.png" />-cobordism theorem or Quinn's thin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094500/t09450032.png" />-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>