Namespaces
Variants
Actions

Difference between revisions of "Two-dimensional annulus"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
t0945001.png
 +
$#A+1 = 29 n = 6
 +
$#C+1 = 29 : ~/encyclopedia/old_files/data/T094/T.0904500 Two\AAhdimensional annulus
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''in topology''
 
''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.
 
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====
 
====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
+
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
  
<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>
+
$$
 +
B  ^ {n} \setminus  h(  \mathop{\rm Int} ( B  ^ {n} ))
 +
$$
  
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" />.
+
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 <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 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 <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 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 <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.
+
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====
 
====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>
 
<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>

Latest revision as of 08:26, 6 June 2020


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 $ 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 $, [a6]; $ n= 3 $,

$ n \geq 5 $,

[a3]; and, finally, $ n= 4 $, [a2], as an application of a special controlled $ h $- cobordism theorem in dimension $ 5 $, called the thin $ h $- cobordism theorem or Quinn's thin $ h $- 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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-dimensional_annulus&oldid=49050
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article