Degree of a mapping

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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 .


[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 map germ" Ann. of Math. , 106 : 1 (1977) pp. 19–38
[5] A.H. Wallace, "Differential topology. First Steps" , Benjamin (1968)
How to Cite This Entry:
Degree of a mapping. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Khokhlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article