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Difference between revisions of "Pontryagin invariant"

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be a closed orientable surface with an  $  n $-
 
be a closed orientable surface with an  $  n $-
 
dimensional framing  $  U $
 
dimensional framing  $  U $
in  $  S  ^ {n+} 2 $,  
+
in  $  S  ^ {n+2} $,  
 
i.e. a trivialization of the normal  $  n $-
 
i.e. a trivialization of the normal  $  n $-
 
dimensional [[Vector bundle|vector bundle]] of the surface  $  M  ^ {2} $
 
dimensional [[Vector bundle|vector bundle]] of the surface  $  M  ^ {2} $
in  $  S  ^ {n+} 2 $.  
+
in  $  S  ^ {n+2} $.  
 
Any element  $  z \in H _ {1} ( M  ^ {2} , \mathbf Z ) $
 
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} $
 
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} $
Line 24: Line 24:
 
restricted to the point  $  f ( y) $,  
 
restricted to the point  $  f ( y) $,  
 
$  y \in C $;  
 
$  y \in C $;  
let  $  u _ {n+} 2 ( y) $
+
let  $  u _ {n+2} ( y) $
 
be the tangent vector to the curve  $  C = f ( S  ^ {1} ) $
 
be the tangent vector to the curve  $  C = f ( S  ^ {1} ) $
 
at the point  $  f ( y) $
 
at the point  $  f ( y) $
 
with respect to the chosen orientation of  $  S  ^ {1} $;  
 
with respect to the chosen orientation of  $  S  ^ {1} $;  
and let  $  u _ {n+} 1 ( y) $
+
and let  $  u _ {n+1} ( y) $
 
be the tangent vector to  $  M  ^ {2} $
 
be the tangent vector to  $  M  ^ {2} $
 
at  $  f ( y) $
 
at  $  f ( y) $
orthogonal to  $  u _ {n+} 2 ( 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) $
+
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 $.  
+
gives the standard orientation of the sphere  $  S  ^ {n+2} $.  
The mapping  $  h :  S  ^ {1} \rightarrow  \mathop{\rm SO} _ {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 ) $(
+
thus arising defines an element of the group  $  \pi _ {1} (  \mathop{\rm SO} _ {n+2} ) $(
 
which is isomorphic to  $  \mathbf Z _ {2} $
 
which is isomorphic to  $  \mathbf Z _ {2} $
 
for  $  n \geq  1 $).  
 
for  $  n \geq  1 $).  
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is zero (Pontryagin's theorem). The Pontryagin invariant can be realized by an  $  ( n + 2 ) $-
 
is zero (Pontryagin's theorem). The Pontryagin invariant can be realized by an  $  ( n + 2 ) $-
 
dimensional framing of the torus,  $  n \geq  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} $,  
+
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 $.
 
$  n \geq  2 $.
  

Revision as of 18:34, 29 December 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