| Version 3 |
Version 2 |
| \theoremstyle{definition} |
\theoremstyle{definition} |
| \newtheorem{defn}{Definition} |
\newtheorem{defn}{Definition} |
|
|
| Suppose that $M$ and $N$ are two oriented differentiable manifolds |
Suppose that $M$ and $N$ are two oriented differentiable manifolds |
| of dimension $n$ (without boundary) with $M$ compact and $N$ connected and suppose that |
of dimension $n$ (without boundary) with $M$ compact and $N$ connected and suppose that |
| $f \colon M \to N$ is a differentiable mapping. Let $Df(x)$ denote the |
$f \colon M \to N$ is a differentiable mapping. Let $Df(x)$ denote the |
| differential mapping at the point $x \in M$, |
differential mapping at the point $x \in M$, |
| that is the linear mapping $Df(x) \colon T_x(M) \to T_{f(x)}(N)$. Let $\operatorname{sign} Df(x)$ denote the sign |
that is the linear mapping $Df(x) \colon T_x(M) \to T_{f(x)}(N)$. Let $\operatorname{sign} Df(x)$ denote the sign |
| of the determinant of $Df(x)$. That is the sign is positive if $f$ preserves |
of the determinant of $Df(x)$. That is the sign is positive if $f$ preserves |
| orientation and negative if $f$ reverses orientation. |
orientation and negative if $f$ reverses orientation. |
|
|
| \begin{defn} |
\begin{defn} |
| Let $y \in N$ be a regular value, then we define the {\em Brower degree} (or just |
Let $y \in N$ be a regular value, then we define the {\em Brower degree} (or just |
| degree) of $f$ by |
degree) of $f$ by |
| \begin{equation*} |
\begin{equation*} |
|
\operatorname{deg} f := \sum_{x \in f^{-1}(y)} \operatorname{sign} Df(x) .
|
\operatorname{deg} f = \sum_{x \in f^{-1}(y)} \operatorname{sign} Df(x) .
|
| \end{equation*} |
\end{equation*} |
| \end{defn} |
\end{defn} |
|
|
| It can be shown that the degree does not depend on the regular value $y$ that we pick so that $\operatorname{deg} f$ is well defined. |
It can be shown that the degree does not depend on the regular value $y$ that we pick so that $\operatorname{deg} f$ is well defined. |
|
|
| Note that this degree coincides with the \PMlinkname{degree}{Degree5} as defined for maps of spheres. |
Note that this degree coincides with the \PMlinkname{degree}{Degree5} as defined for maps of spheres. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Milnor:topdiff} |
\bibitem{Milnor:topdiff} |
| John~W. Milnor. |
John~W. Milnor. |
| {\em \PMlinkescapetext{Topology From The Differentiable Viewpoint}}. |
{\em \PMlinkescapetext{Topology From The Differentiable Viewpoint}}. |
| The University Press of Virginia, Charlottesville, Virginia, 1969. |
The University Press of Virginia, Charlottesville, Virginia, 1969. |
| \end{thebibliography} |
\end{thebibliography} |
