From Sn to Sn The simplest and most important case is the degree of a
continuous map from the
n-sphere S^n to itself (in the case n=1, this is called the winding number): Let f\colon S^n\to S^n be a continuous map. Then f induces a
pushforward homomorphism f_*\colon H_n\left(S^n\right) \to H_n\left(S^n\right), where H_n\left(\cdot\right) is the nth
homology group. Considering the fact that H_n\left(S^n\right)\cong\mathbb{Z}, we see that f_* must be of the form f_*\colon x\mapsto\alpha x for some fixed \alpha\in\mathbb{Z}. This \alpha is then called the degree of f.
Between manifolds Algebraic topology Let
X and
Y be closed
connected oriented m-dimensional
manifolds.
Poincare duality implies that the manifold's top
homology group is isomorphic to
Z. Choosing an orientation means choosing a generator of the top homology group. A continuous map
f :
X →
Y induces a homomorphism
f∗ from
Hm(
X) to
Hm(
Y). Let [
X], resp. [
Y] be the chosen generator of
Hm(
X), resp.
Hm(
Y) (or the
fundamental class of
X,
Y). Then the
degree of
f is defined to be
f∗([
X]). In other words, :f_*([X]) = \deg(f)[Y] \, . If
y in
Y and
f −1(
y) is a finite set, the degree of
f can be computed by considering the
m-th
local homology groups of
X at each point in
f −1(
y). Namely, if f^{-1}(y)=\{x_1,\dots,x_m\}, then :\deg(f) = \sum_{i=1}^{m}\deg(f|_{x_i}) \, .
Differential topology In the language of
differential topology, the degree of a smooth map can be defined as follows: If
f is a smooth map whose domain is a compact manifold and
p is a
regular value of
f, consider the finite set :f^{-1}(p) = \{x_1, x_2, \ldots, x_n\} \,. By
p being a regular value, in a neighborhood of each
xi the map
f is a local
diffeomorphism. Diffeomorphisms can be either orientation preserving or orientation reversing. Let
r be the number of points
xi at which
f is orientation preserving and
s be the number at which
f is orientation reversing. When the codomain of
f is connected, the number
r −
s is independent of the choice of
p (though
n is not!) and one defines the
degree of
f to be
r −
s. This definition coincides with the algebraic topological definition above. The same definition works for compact manifolds with
boundary but then
f should send the boundary of
X to the boundary of
Y. One can also define
degree modulo 2 (deg2(
f)) the same way as before but taking the
fundamental class in
Z2 homology. In this case deg2(
f) is an element of
Z2 (the
field with two elements), the manifolds need not be orientable and if
n is the number of preimages of
p as before then deg2(
f) is
n modulo 2. Integration of
differential forms gives a pairing between (C∞-)
singular homology and
de Rham cohomology: \langle c, \omega\rangle = \int_c \omega, where c is a homology class represented by a cycle c and \omega a closed form representing a de Rham cohomology class. For a smooth map
f:
X →
Y between orientable
m-manifolds, one has :\left\langle f_* [c], [\omega] \right\rangle = \left\langle [c], f^*[\omega] \right\rangle, where
f∗ and
f∗ are induced maps on chains and forms respectively. Since
f∗[
X] = deg
f · [
Y], we have :\deg f \int_Y \omega = \int_X f^*\omega \, for any
m-form
ω on
Y.
Maps from closed region If \Omega \subset \R^n is a bounded
region, f: \bar\Omega \to \R^n smooth, p a
regular value of f and p \notin f(\partial\Omega), then the degree \deg(f, \Omega, p) is defined by the formula :\deg(f, \Omega, p) := \sum_{y\in f^{-1}(p)} \sgn \det(Df(y)) where Df(y) is the
Jacobian matrix of f in y. This definition of the degree may be naturally extended for non-regular values p such that \deg(f, \Omega, p) = \deg\left(f, \Omega, p'\right) where p' is a point close to p. The topological degree can also be calculated using a
surface integral over the boundary of \Omega, and if \Omega is a connected
n-
polytope, then the degree can be expressed as a sum of determinants over a certain subdivision of its
facets. The degree satisfies the following properties: • If \deg\left(f, \bar\Omega, p\right) \neq 0, then there exists x \in \Omega such that f(x) = p. • \deg(\operatorname{id}, \Omega, y) = 1 for all y \in \Omega. • Decomposition property: \deg(f, \Omega, y) = \deg(f, \Omega_1, y) + \deg(f, \Omega_2, y), if \Omega_1, \Omega_2 are disjoint parts of \Omega = \Omega_1 \cup \Omega_2 and y \not\in f{\left(\overline{\Omega}\setminus\left(\Omega_1 \cup \Omega_2\right)\right)}. •
Homotopy invariance: If f and g are
homotopy equivalent via a homotopy F(t) such that F(0) = f,\, F(1) = g and p \notin F(t)(\partial\Omega), then \deg(f, \Omega, p) = \deg(g, \Omega, p). • The function p \mapsto \deg(f, \Omega, p) is locally constant on \R^n - f(\partial\Omega). These properties characterise the degree uniquely and the degree may be defined by them in an axiomatic way. In a similar way, we could define the degree of a map between compact oriented
manifolds with boundary. ==Properties==