Difference between revisions of "Degree of a mapping"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1980) {{MR|0606196}} {{ZBL|0434.55001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W. Milnor, "Toplogy from the differentiable viewpoint" , Univ. Virginia Press (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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) {{MR|777682}} {{ZBL|0554.58001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Eisenbud, H. Levine, "An algebraic formula for the degree of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090040.png" /> map germ" ''Ann. of Math.'' , '''106''' : 1 (1977) pp. 19–38 {{MR|467800}} {{ZBL|0398.57020}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.H. Wallace, "Differential topology. First Steps" , Benjamin (1968) {{MR|0436148}} {{MR|0224103}} {{ZBL|0164.23805}} </TD></TR></table> |
Revision as of 16:56, 15 April 2012
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) MR0606196 Zbl 0434.55001 |
[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) MR777682 Zbl 0554.58001 |
[4] | D. Eisenbud, H. Levine, "An algebraic formula for the degree of a ![]() |
[5] | A.H. Wallace, "Differential topology. First Steps" , Benjamin (1968) MR0436148 MR0224103 Zbl 0164.23805 |
Degree of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degree_of_a_mapping&oldid=14689