Pontryagin invariant
An invariant of framed constructions of surfaces with a given framing. Let be a closed orientable surface with an
-dimensional framing
in
, i.e. a trivialization of the normal
-dimensional vector bundle of the surface
in
. Any element
can be realized by a smoothly immersed circle with self-intersections which are only double points and transversal. Let some orientation of the circle
be fixed; let
be orthogonal vectors from
restricted to the point
,
; let
be the tangent vector to the curve
at the point
with respect to the chosen orientation of
; and let
be the tangent vector to
at
orthogonal to
and oriented such that the sequence of vectors
gives the standard orientation of the sphere
. The mapping
thus arising defines an element of the group
(which is isomorphic to
for
). Let
if
is homotopic to zero and
if
is not homotopic to zero. Let the value of the function
be equal to the sum modulo 2 of the number of double points of the curve
realizing the element
and the number
defined by
. Thus, a given value of
depends only on the homology class of
, and the function
satisfies the following condition:
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where is the intersection form of one-dimensional homologies of the surface
. The Arf-invariant of
is called the Pontryagin invariant of the pair
. The pair
admits a framed surgery to the pair
if and only if the Pontryagin invariant of the pair
is zero (Pontryagin's theorem). The Pontryagin invariant can be realized by an
-dimensional framing of the torus,
, and is the unique invariant of two-dimensional framed cobordism. The Pontryagin invariant defines an isomorphism
,
.
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) |
Pontryagin invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_invariant&oldid=48241