From Sⁿ to Sⁿ

The simplest and most important case is the degree of a continuous map from the ${\displaystyle n}$-sphere ${\displaystyle S^{n}}$ to itself (in the case ${\displaystyle n=1}$, this is called the winding number):

Let ${\displaystyle f\colon S^{n}\to S^{n}}$ be a continuous map. Then ${\displaystyle f}$ induces a homomorphism ${\displaystyle f_{*}\colon H_{n}\left(S^{n}\right)\to H_{n}\left(S^{n}\right)}$, where ${\displaystyle H_{n}\left(\cdot \right)}$ is the ${\displaystyle n}$th homology group. Considering the fact that ${\displaystyle H_{n}\left(S^{n}\right)\cong \mathbb {Z} }$, we see that ${\displaystyle f_{*}}$ must be of the form ${\displaystyle f_{*}\colon x\mapsto \alpha x}$ for some fixed ${\displaystyle \alpha \in \mathbb {Z} }$. This ${\displaystyle \alpha }$ is then called the degree of ${\displaystyle f}$.

Between manifolds

Maps from closed region

If ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$is a bounded region, ${\displaystyle f:{\bar {\Omega }}\to \mathbb {R} ^{n}}$ smooth, ${\displaystyle p}$ a regular value of ${\displaystyle f}$ and ${\displaystyle p\notin f(\partial \Omega )}$, then the degree ${\displaystyle \deg(f,\Omega ,p)}$ is defined by the formula

${\displaystyle \deg(f,\Omega ,p):=\sum _{y\in f^{-1}(p)}\operatorname {sgn} \det(Df(y))}$

where ${\displaystyle Df(y)}$ is the Jacobian matrix of ${\displaystyle f}$ in ${\displaystyle y}$.

This definition of the degree may be naturally extended for non-regular values ${\displaystyle p}$ such that ${\displaystyle \deg(f,\Omega ,p)=\deg \left(f,\Omega ,p'\right)}$ where ${\displaystyle p'}$ is a point close to ${\displaystyle p}$.

The degree satisfies the following properties:^[2]

• If ${\displaystyle \deg \left(f,{\bar {\Omega }},p\right)\neq 0}$, then there exists ${\displaystyle x\in \Omega }$ such that ${\displaystyle f(x)=p}$.
• ${\displaystyle \deg(\operatorname {id} ,\Omega ,y)=1}$ for all ${\displaystyle y\in \Omega }$.
• Decomposition property:
  $${\displaystyle \deg(f,\Omega ,y)=\deg(f,\Omega _{1},y)+\deg(f,\Omega _{2},y),}$$

  
  if ${\displaystyle \Omega _{1},\Omega _{2}}$ are disjoint parts of ${\displaystyle \Omega =\Omega _{1}\cup \Omega _{2}}$ and ${\displaystyle y\not \in f{\left({\overline {\Omega }}\setminus \left(\Omega _{1}\cup \Omega _{2}\right)\right)}}$.
• Homotopy invariance: If ${\displaystyle f}$ and ${\displaystyle g}$ are homotopy equivalent via a homotopy ${\displaystyle F(t)}$ such that ${\displaystyle F(0)=f,\,F(1)=g}$ and ${\displaystyle p\notin F(t)(\partial \Omega )}$, then ${\displaystyle \deg(f,\Omega ,p)=\deg(g,\Omega ,p)}$
• The function ${\displaystyle p\mapsto \deg(f,\Omega ,p)}$ is locally constant on ${\displaystyle \mathbb {R} ^{n}-f(\partial \Omega )}$

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.