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An invariant of framed constructions of surfaces with a given framing. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p0737701.png" /> be a closed orientable surface with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p0737702.png" />-dimensional framing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p0737703.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p0737704.png" />, i.e. a trivialization of the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p0737705.png" />-dimensional [[Vector bundle|vector bundle]] of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p0737706.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p0737707.png" />. Any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p0737708.png" /> can be realized by a smoothly immersed circle with self-intersections which are only double points and transversal. Let some orientation of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p0737709.png" /> be fixed; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377010.png" /> be orthogonal vectors from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377011.png" /> restricted to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377013.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377014.png" /> be the tangent vector to the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377015.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377016.png" /> with respect to the chosen orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377017.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377018.png" /> be the tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377019.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377020.png" /> orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377021.png" /> and oriented such that the sequence of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377022.png" /> gives the standard orientation of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377023.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377024.png" /> thus arising defines an element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377025.png" /> (which is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377027.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377028.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377029.png" /> is homotopic to zero and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377030.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377031.png" /> is not homotopic to zero. Let the value of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377032.png" /> be equal to the sum modulo 2 of the number of double points of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377033.png" /> realizing the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377034.png" /> and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377035.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377036.png" />. Thus, a given value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377037.png" /> depends only on the homology class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377038.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377039.png" /> satisfies the following condition:
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<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/p/p073/p073770/p07377040.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377041.png" /> is the intersection form of one-dimensional homologies of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377042.png" />. The [[Arf-invariant|Arf-invariant]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377043.png" /> is called the Pontryagin invariant of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377044.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377045.png" /> admits a framed surgery to the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377046.png" /> if and only if the Pontryagin invariant of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377047.png" /> is zero (Pontryagin's theorem). The Pontryagin invariant can be realized by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377048.png" />-dimensional framing of the torus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377049.png" />, and is the unique invariant of two-dimensional framed [[Cobordism|cobordism]]. The Pontryagin invariant defines an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073770/p07377051.png" />.
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An invariant of framed constructions of surfaces with a given framing. Let  $  ( M  ^ {2} , U ) $
 +
be a closed orientable surface with an  $  n $-
 +
dimensional framing  $  U $
 +
in  $  S  ^ {n+} 2 $,
 +
i.e. a trivialization of the normal  $  n $-
 +
dimensional [[Vector bundle|vector bundle]] of the surface  $  M  ^ {2} $
 +
in  $  S  ^ {n+} 2 $.
 +
Any element  $  z \in H _ {1} ( M  ^ {2} , \mathbf Z ) $
 +
can be realized by a smoothly immersed circle with self-intersections which are only double points and transversal. Let some orientation of the circle  $  S  ^ {1} $
 +
be fixed; let  $  u _ {1} ( y) \dots u _ {n} ( y) $
 +
be orthogonal vectors from  $  U $
 +
restricted to the point  $  f ( y) $,
 +
$  y \in C $;
 +
let  $  u _ {n+} 2 ( y) $
 +
be the tangent vector to the curve  $  C = f ( S  ^ {1} ) $
 +
at the point  $  f ( y) $
 +
with respect to the chosen orientation of  $  S  ^ {1} $;
 +
and let  $  u _ {n+} 1 ( y) $
 +
be the tangent vector to  $  M  ^ {2} $
 +
at  $  f ( y) $
 +
orthogonal to  $  u _ {n+} 2 ( y) $
 +
and oriented such that the sequence of vectors  $  u _ {1} ( y) \dots u _ {n} ( y) , u _ {n+} 1 ( y) , u _ {n+} 2 ( y) $
 +
gives the standard orientation of the sphere  $  S  ^ {n+} 2 $.
 +
The mapping  $  h :  S  ^ {1} \rightarrow  \mathop{\rm SO} _ {n+} 2 $
 +
thus arising defines an element of the group  $  \pi _ {1} (  \mathop{\rm SO} _ {n+} 2 ) $(
 +
which is isomorphic to  $  \mathbf Z _ {2} $
 +
for  $  n \geq  1 $).
 +
Let  $  \beta = 0 $
 +
if  $  h $
 +
is homotopic to zero and  $  \beta = 1 $
 +
if  $  h $
 +
is not homotopic to zero. Let the value of the function  $  \Phi _ {0} : H _ {1} ( M  ^ {2} , \mathbf Z ) \rightarrow \mathbf Z _ {2} $
 +
be equal to the sum modulo 2 of the number of double points of the curve  $  C $
 +
realizing the element  $  z $
 +
and the number  $  \beta $
 +
defined by  $  C $.  
 +
Thus, a given value of  $  \Phi _ {0} ( z) $
 +
depends only on the homology class of  $  z $,
 +
and the function  $  \Phi _ {0} ( z) $
 +
satisfies the following condition:
 +
 
 +
$$
 +
\Phi _ {0} ( z _ {1} + z _ {2} )  =  \Phi _ {0} ( z _ {1} ) +
 +
\Phi _ {0} ( z _ {2} ) + \Phi ( z _ {1} , z _ {2} ) \  \mathop{\rm mod}  2 ,
 +
$$
 +
 
 +
where  $  \Phi :  H _ {1} ( M  ^ {2} , \mathbf Z ) \times H _ {1} ( M  ^ {2} , \mathbf Z ) \rightarrow \mathbf Z $
 +
is the intersection form of one-dimensional homologies of the surface $  M  ^ {2} $.  
 +
The [[Arf-invariant|Arf-invariant]] of $  \Phi _ {0} $
 +
is called the Pontryagin invariant of the pair $  ( M  ^ {2} , U ) $.  
 +
The pair $  ( M  ^ {2} , U ) $
 +
admits a framed surgery to the pair $  ( S  ^ {2} , U ) $
 +
if and only if the Pontryagin invariant of the pair $  ( M  ^ {2} , U ) $
 +
is zero (Pontryagin's theorem). The Pontryagin invariant can be realized by an $  ( n + 2 ) $-
 +
dimensional framing of the torus, $  n \geq  2 $,  
 +
and is the unique invariant of two-dimensional framed [[Cobordism|cobordism]]. The Pontryagin invariant defines an isomorphism $  \pi _ {n+} 2 ( S  ^ {n} ) \approx \mathbf Z _ {2} $,  
 +
$  n \geq  2 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Smooth manifolds and their applications in homology theory" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Smooth manifolds and their applications in homology theory" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.E. Stong,  "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.W. Milnor,  "Toplogy from the differentiable viewpoint" , Univ. Virginia Press  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.E. Stong,  "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.W. Milnor,  "Toplogy from the differentiable viewpoint" , Univ. Virginia Press  (1966)</TD></TR></table>

Revision as of 08:07, 6 June 2020


An invariant of framed constructions of surfaces with a given framing. Let $ ( M ^ {2} , U ) $ be a closed orientable surface with an $ n $- dimensional framing $ U $ in $ S ^ {n+} 2 $, i.e. a trivialization of the normal $ n $- dimensional vector bundle of the surface $ M ^ {2} $ in $ S ^ {n+} 2 $. Any element $ z \in H _ {1} ( M ^ {2} , \mathbf Z ) $ can be realized by a smoothly immersed circle with self-intersections which are only double points and transversal. Let some orientation of the circle $ S ^ {1} $ be fixed; let $ u _ {1} ( y) \dots u _ {n} ( y) $ be orthogonal vectors from $ U $ restricted to the point $ f ( y) $, $ y \in C $; let $ u _ {n+} 2 ( y) $ be the tangent vector to the curve $ C = f ( S ^ {1} ) $ at the point $ f ( y) $ with respect to the chosen orientation of $ S ^ {1} $; and let $ u _ {n+} 1 ( y) $ be the tangent vector to $ M ^ {2} $ at $ f ( y) $ orthogonal to $ u _ {n+} 2 ( y) $ and oriented such that the sequence of vectors $ u _ {1} ( y) \dots u _ {n} ( y) , u _ {n+} 1 ( y) , u _ {n+} 2 ( y) $ gives the standard orientation of the sphere $ S ^ {n+} 2 $. The mapping $ h : S ^ {1} \rightarrow \mathop{\rm SO} _ {n+} 2 $ thus arising defines an element of the group $ \pi _ {1} ( \mathop{\rm SO} _ {n+} 2 ) $( which is isomorphic to $ \mathbf Z _ {2} $ for $ n \geq 1 $). Let $ \beta = 0 $ if $ h $ is homotopic to zero and $ \beta = 1 $ if $ h $ is not homotopic to zero. Let the value of the function $ \Phi _ {0} : H _ {1} ( M ^ {2} , \mathbf Z ) \rightarrow \mathbf Z _ {2} $ be equal to the sum modulo 2 of the number of double points of the curve $ C $ realizing the element $ z $ and the number $ \beta $ defined by $ C $. Thus, a given value of $ \Phi _ {0} ( z) $ depends only on the homology class of $ z $, and the function $ \Phi _ {0} ( z) $ satisfies the following condition:

$$ \Phi _ {0} ( z _ {1} + z _ {2} ) = \Phi _ {0} ( z _ {1} ) + \Phi _ {0} ( z _ {2} ) + \Phi ( z _ {1} , z _ {2} ) \ \mathop{\rm mod} 2 , $$

where $ \Phi : H _ {1} ( M ^ {2} , \mathbf Z ) \times H _ {1} ( M ^ {2} , \mathbf Z ) \rightarrow \mathbf Z $ is the intersection form of one-dimensional homologies of the surface $ M ^ {2} $. The Arf-invariant of $ \Phi _ {0} $ is called the Pontryagin invariant of the pair $ ( M ^ {2} , U ) $. The pair $ ( M ^ {2} , U ) $ admits a framed surgery to the pair $ ( S ^ {2} , U ) $ if and only if the Pontryagin invariant of the pair $ ( M ^ {2} , U ) $ is zero (Pontryagin's theorem). The Pontryagin invariant can be realized by an $ ( n + 2 ) $- dimensional framing of the torus, $ n \geq 2 $, and is the unique invariant of two-dimensional framed cobordism. The Pontryagin invariant defines an isomorphism $ \pi _ {n+} 2 ( S ^ {n} ) \approx \mathbf Z _ {2} $, $ n \geq 2 $.

References

[1] L.S. Pontryagin, "Smooth manifolds and their applications in homology theory" , Moscow (1976) (In Russian)

Comments

References

[a1] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968)
[a2] J.W. Milnor, "Toplogy from the differentiable viewpoint" , Univ. Virginia Press (1966)
How to Cite This Entry:
Pontryagin invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_invariant&oldid=48241
This article was adapted from an original article by M.A. Shtan'ko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article