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)