# Pontryagin invariant

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) |

[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=53580