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''framed manifold''
 
''framed manifold''
  
A smooth manifold with a fixed trivialization of the [[Normal bundle|normal bundle]]. More accurately, let a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r0823501.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r0823502.png" /> be imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r0823503.png" /> and let the (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r0823504.png" />-dimensional) normal fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r0823505.png" /> corresponding to this imbedding be trivial. Any trivialization of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r0823506.png" /> is called a rigging (framing) of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r0823507.png" /> corresponding to this imbedding. Framed manifolds were introduced around 1950 (see [[#References|[1]]]) in order to prove that the [[Cobordism|cobordism]] groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r0823508.png" />-dimensional framed manifolds lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r0823509.png" /> are isomorphic to the homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235010.png" />; the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235012.png" /> have been computed along these lines.
+
A smooth manifold with a fixed trivialization of the [[Normal bundle|normal bundle]]. More accurately, let a smooth $  n $-
 +
dimensional manifold $  M $
 +
be imbedded in $  \mathbf R  ^ {n+} k $
 +
and let the ( $  k $-
 +
dimensional) normal fibration $  \nu $
 +
corresponding to this imbedding be trivial. Any trivialization of the fibration $  \nu $
 +
is called a rigging (framing) of the manifold $  M $
 +
corresponding to this imbedding. Framed manifolds were introduced around 1950 (see [[#References|[1]]]) in order to prove that the [[Cobordism|cobordism]] groups of $  n $-
 +
dimensional framed manifolds lying in $  \mathbf R  ^ {n+} k $
 +
are isomorphic to the homotopy groups $  \pi _ {n+} k ( S  ^ {n} ) $;  
 +
the groups $  \pi _ {n+} 1 ( S  ^ {n} ) $
 +
and $  \pi _ {n+} 2 ( S  ^ {n} ) $
 +
have been computed along these lines.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Smooth manifolds and their application in homotopy theory"  ''Trudy Mat. Inst. Steklov.'' , '''45'''  (1955)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Smooth manifolds and their application in homotopy theory"  ''Trudy Mat. Inst. Steklov.'' , '''45'''  (1955)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The phrase  "framed manifold"  is also used to denote a differentiable manifold together with a basis in each fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235013.png" /> of its tangent bundle, with the chosen basis depending differentiably on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235014.png" />.
+
The phrase  "framed manifold"  is also used to denote a differentiable manifold together with a basis in each fibre $  T _ {x} M $
 +
of its tangent bundle, with the chosen basis depending differentiably on $  x $.
  
The frame bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235015.png" /> over an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235016.png" />-dimensional smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235017.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235018.png" />-dimensional smooth fibre bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235019.png" /> (so its total space is of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235020.png" />) whose fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235021.png" /> consists of all linear isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235022.png" />. Equivalently, the fibre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235023.png" /> consists of all ordered bases, also called frames, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235024.png" />. Thus, more precisely, a framed manifold is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235025.png" /> consisting of a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235026.png" /> together with a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082350/r08235027.png" /> of the frame bundle. Such a section is called a framing.
+
The frame bundle $  F( M) $
 +
over an $  n $-
 +
dimensional smooth manifold $  M $
 +
is the $  n  ^ {2} $-
 +
dimensional smooth fibre bundle over $  M $(
 +
so its total space is of dimension $  n  ^ {2} + n $)  
 +
whose fibre over $  x \in M $
 +
consists of all linear isomorphisms $  T _ {x} M \simeq \mathbf R  ^ {n} $.  
 +
Equivalently, the fibre at $  x $
 +
consists of all ordered bases, also called frames, for $  T _ {x} M $.  
 +
Thus, more precisely, a framed manifold is a pair $  ( M , s) $
 +
consisting of a smooth manifold $  M $
 +
together with a section $  s : M \rightarrow F ( M) $
 +
of the frame bundle. Such a section is called a framing.
  
 
Quite generally, of course, the word  "frame"  is used as a substitute for basis in a vector space. The terminology derives from the fact that a basis in space-time provides a frame of reference in the sense of mechanics.
 
Quite generally, of course, the word  "frame"  is used as a substitute for basis in a vector space. The terminology derives from the fact that a basis in space-time provides a frame of reference in the sense of mechanics.

Revision as of 08:11, 6 June 2020


framed manifold

A smooth manifold with a fixed trivialization of the normal bundle. More accurately, let a smooth $ n $- dimensional manifold $ M $ be imbedded in $ \mathbf R ^ {n+} k $ and let the ( $ k $- dimensional) normal fibration $ \nu $ corresponding to this imbedding be trivial. Any trivialization of the fibration $ \nu $ is called a rigging (framing) of the manifold $ M $ corresponding to this imbedding. Framed manifolds were introduced around 1950 (see [1]) in order to prove that the cobordism groups of $ n $- dimensional framed manifolds lying in $ \mathbf R ^ {n+} k $ are isomorphic to the homotopy groups $ \pi _ {n+} k ( S ^ {n} ) $; the groups $ \pi _ {n+} 1 ( S ^ {n} ) $ and $ \pi _ {n+} 2 ( S ^ {n} ) $ have been computed along these lines.

References

[1] L.S. Pontryagin, "Smooth manifolds and their application in homotopy theory" Trudy Mat. Inst. Steklov. , 45 (1955) (In Russian)

Comments

The phrase "framed manifold" is also used to denote a differentiable manifold together with a basis in each fibre $ T _ {x} M $ of its tangent bundle, with the chosen basis depending differentiably on $ x $.

The frame bundle $ F( M) $ over an $ n $- dimensional smooth manifold $ M $ is the $ n ^ {2} $- dimensional smooth fibre bundle over $ M $( so its total space is of dimension $ n ^ {2} + n $) whose fibre over $ x \in M $ consists of all linear isomorphisms $ T _ {x} M \simeq \mathbf R ^ {n} $. Equivalently, the fibre at $ x $ consists of all ordered bases, also called frames, for $ T _ {x} M $. Thus, more precisely, a framed manifold is a pair $ ( M , s) $ consisting of a smooth manifold $ M $ together with a section $ s : M \rightarrow F ( M) $ of the frame bundle. Such a section is called a framing.

Quite generally, of course, the word "frame" is used as a substitute for basis in a vector space. The terminology derives from the fact that a basis in space-time provides a frame of reference in the sense of mechanics.

References

[a1] J.W. Milnor, "Toplogy from the differentiable viewpoint" , Univ. Virginia Press (1965)
[a2] J.W. Milnor, "A survey of cobordism theory" L'Enseign. Math. , 8 (1962) pp. 16–23
[a3] R. Thom, "Quelque propriétés globales des variétés différentiables" Comm. Math. Helvet. , 28 (1954) pp. 17–28
[a4] J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)
[a5] C.T.J. Dodson, "Categories, bundles, and spacetime topology" , Kluwer (1988) pp. 94ff
[a6] M.W. Hirsch, "Differential topology" , Springer (1976) pp. 98
How to Cite This Entry:
Rigged manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rigged_manifold&oldid=48571
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article