# 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
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