nLab degree of a continuous function

Idea

Given a continuous function between two connected closed oriented topological 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 $f \colon X \to Y$ between two such manifolds, the homomorphism $f_\ast = H_n(f) \colon H_n(X) \to H_n(Y)$ is therefore specified by the integer $n$ such that $f_\ast [\omega_X] = n [\omega_Y]$. 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 $g \colon X \to Y$ is frequently computed according to the following considerations:

• The space of continuous functions $g \colon X \to Y$ has a dense subspace consisting of smooth functions $f \colon X \to Y$, 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 $x_1, \ldots, x_r \in X$. Since these are regular points, $f$ restricts to a diffeomorphism

  $$f_i \colon U_i \to V$$

  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

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

  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:

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

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

Properties

Hopf degree theorem

Proposition

(Hopf degree theorem)

Let $n \in \mathbb{N}$ be a natural number and $X \in Mfd$ be a connected orientable closed manifold of dimension $n$. Then the $n$th cohomotopy classes $\left[X \overset{c}{\to} S^n\right] \in \pi^n(X)$ of $X$ are in bijection to the degree $deg(c) \in \mathbb{Z}$ of the representing functions, hence the canonical function

$$\pi^n(X) \underoverset{\simeq}{S^n \to K(\mathbb{Z},n)}{\longrightarrow} H^n(X,\mathbb{Z}) \;\simeq\; \mathbb{Z}$$

from $n$th cohomotopy to $n$th integral cohomology is a bijection.

(e.g. Kosinski 93, IX (5.8), Kobin 16, 7.5)

Poincaré–Hopf theorem

See at Poincaré–Hopf theorem.

Generalization to the Adams d-invariant

The Hopf degree of a map is a special case of its Adams d-invariant; see there for more.

Examples

• winding number

Related concepts

• Poincaré–Hopf theorem

References

Texbook accounts:

• B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, section 13 of: Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985

See also:

• Wikipedia, Degree of a continuous mapping