Degree of a mapping
degree of a continuous mapping between connected compact manifolds of identical dimension
An integer such that
, where
are the fundamental classes (cf. Fundamental class) of the manifolds
and
over the ring
or
, and
is the induced mapping. In the case of non-orientable manifolds, the degree of the mapping is uniquely defined modulo 2. If
is a differentiable mapping between closed differentiable manifolds, then
modulo 2 coincides with the number of inverse images of a regular value
of
. In the case of oriented manifolds
![]() |
where is the sign of the Jacobian of
at a point
(the Browder degree).
For a continuous mapping and an isolated point
in the inverse image of zero, the concept of the local degree
at the point
is defined:
, where
is the restriction of
onto a small sphere
![]() |
and is the projection from zero onto the unit sphere. In the case of a differentiable
, the formula
![]() |
holds, where is the ring of germs (cf. Germ) of smooth functions at zero, factorized by the ideal generated by the components of
, and
is the maximal ideal of the quotient ring relative to the property
. Let
be the class of the Jacobian of the mapping
. Then for a linear form
such that
the formula
holds, where
is a symmetric bilinear form on
.
References
[1] | A. Dold, "Lectures on algebraic topology" , Springer (1980) |
[2] | J.W. Milnor, "Toplogy from the differentiable viewpoint" , Univ. Virginia Press (1965) |
[3] | V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , 1 , Birkhäuser (1985) (Translated from Russian) |
[4] | D. Eisenbud, H. Levine, "An algebraic formula for the degree of a ![]() |
[5] | A.H. Wallace, "Differential topology. First Steps" , Benjamin (1968) |
Degree of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degree_of_a_mapping&oldid=14689