# Degree of a mapping

degree of a continuous mapping $f: ( M, \partial M) \rightarrow ( N, \partial N)$ between connected compact manifolds of identical dimension

An integer $\mathop{\rm deg} f$ such that $f _ \star ( \mu _ {M} ) = \mathop{\rm deg} f \cdot \mu _ {N}$, where $\mu _ {M} , \mu _ {N}$ are the fundamental classes (cf. Fundamental class) of the manifolds $M$ and $N$ over the ring $\mathbf Z$ or $\mathbf Z _ {2}$, and $f _ \star$ is the induced mapping. In the case of non-orientable manifolds, the degree of the mapping is uniquely defined modulo 2. If $f: M \rightarrow N$ is a differentiable mapping between closed differentiable manifolds, then $\mathop{\rm deg} f$ modulo 2 coincides with the number of inverse images of a regular value $y$ of $f$. In the case of oriented manifolds

$$\mathop{\rm deg} f = \sum _ {x \in f ^ {-} 1 ( y) } \mathop{\rm sign} J _ {x} ,$$

where $\mathop{\rm sign} J _ {x}$ is the sign of the Jacobian of $f$ at a point $x$( the Browder degree).

For a continuous mapping $f: ( \mathbf R ^ {n} , 0) \rightarrow ( \mathbf R ^ {n} , 0)$ and an isolated point $x$ in the inverse image of zero, the concept of the local degree $\mathop{\rm deg} _ {x} f$ at the point $x$ is defined: $\mathop{\rm deg} _ {x} f = \mathop{\rm deg} \pi \circ h$, where $h$ is the restriction of $f$ onto a small sphere

$$S _ \epsilon ^ {n} = \partial B _ \epsilon ^ {n} ,\ \ B _ \epsilon ^ {n} \cap f ^ { - 1 } ( 0) = \ x \in \mathop{\rm Int} B _ \epsilon ^ {n} ,$$

and $\pi$ is the projection from zero onto the unit sphere. In the case of a differentiable $f$, the formula

$$| \mathop{\rm deg} _ {x} f | = \mathop{\rm dim} Q( f ) - 2 \mathop{\rm dim} I$$

holds, where $Q( f )$ is the ring of germs (cf. Germ) of smooth functions at zero, factorized by the ideal generated by the components of $f$, and $I$ is the maximal ideal of the quotient ring relative to the property $I ^ {2} = 0$. Let $J _ {0} \in Q( f )$ be the class of the Jacobian of the mapping $f$. Then for a linear form $\phi : Q( f ) \rightarrow \mathbf R$ such that $\phi ( J _ {0} ) > 0$ the formula $\mathop{\rm deg} _ {x} f = \mathop{\rm Index} \langle , \rangle _ \phi$ holds, where $\langle p, q\rangle _ \phi = \phi ( p \cdot q)$ is a symmetric bilinear form on $Q( f )$.

#### 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 map germ" Ann. of Math. , 106 : 1 (1977) pp. 19–38 MR467800 Zbl 0398.57020 [5] A.H. Wallace, "Differential topology. First Steps" , Benjamin (1968) MR0436148 MR0224103 Zbl 0164.23805
How to Cite This Entry:
Degree of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degree_of_a_mapping&oldid=46618
This article was adapted from an original article by A.V. Khokhlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article