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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====
<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====

====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>