nLab
degree of a continuous function

Idea

Given a continuous function between two connected closed oriented manifolds of the same dimension, its degree is a measure for how often the function “wraps its domain around its codomain”.

Definition

For $X$ is a connected closed oriented manifold of dimension $n$, its top homology group $H_n(X)=H_n(X;\mathbb{Z})$ is isomorphic to $\mathbb{Z}$, where the generator $1\in \mathbb{Z}$ is identified with the orientation class $[{\omega }_X]$ of $X$, the fundamental class of $X$.

Definition

Given a continuous map $f:X\to Y$ between two such manifolds, the homomorphism $f_{*}=H_n(f):H_n(X)\to H_n(Y)$ is therefore specified by the integer $n$ such that $f_{*}[{\omega }_X]=n[{\omega }_Y]$. This integer is called the degree of $f$.

Computing the degree

We suppose throughout that $X$ and $Y$ are connected closed oriented manifolds of the same dimension $n$. The degree of a continuous function $g:X\to Y$ is frequently computed according to the following considerations:

• The space of continuous functions $g:X\to Y$ has a dense subspace consisting of smooth functions $f:X\to Y$, and in particular every continuous function $g$ is homotopic to a smooth function $f$. It therefore suffices to compute the degree of $f$.

• By Sard's theorem?, the set of singular values? of a smooth function $f$ has measure zero (using for example the orientation on $Y$ to define a volume form and hence a measure). Accordingly, we may choose a regular value? $y\in Y$.

• The inverse image $f^{-1}(y)$ is a compact $0$-dimensional manifold, hence consists of finitely many (possibly zero) points $x_1,\dots ,x_r\in X$. Since these are regular points, $f$ restricts to a diffeomorphism

  $$f_i:U_i\to V$$f_i \colon U_i \to V

  where $U_i$ is a small neighborhood of $x_i$ and $V$ is a small neighborhood of $y$. The diffeomorphism $f_i$ either preserves or reverses the orientation of $U_i$, i.e., the sign of the determinant as a mapping between differential n-forms

  $${\Omega }^n(U_i)\to {\Omega }^n(V)$$\Omega^n(U_i) \to \Omega^n(V)

  is either $+1$ or $-1$.

• By a straightforward application of the excision axiom? in homology, it follows that the degree of $f$ is the sum of these signs:

  $$\mathrm{deg}(f)=\sum _{i=1}^r\mathrm{sign}({\Omega }^n(f_i))$$\deg(f) = \sum_{i=1}^r sign(\Omega^n(f_i))

  and this quantity is independent of the choice of regular value $y$.

References

• Wikipedia, Degree of a continuous mapping