Brouwer’s Topological Degree

We present here a series of posts devoted to Brouwer’s topological degree. Our aim is to motivate the study of a poorly-known yet powerful tool in the study of topological properties of $\mathbb{R}^n$ ; in particular, it will be shown that many theorems, like Jordan curve theorem or invariance of domain theorem or hairy ball theorem, usually proved using homology, can also be proved using Brouwer’s topological degree, while homology needs more work to be established.

The problem is the following: let $\Omega \subset \mathbb{R}^n$ be an open subspace, $f : \Omega \to \mathbb{R}^n$ be a continuous map and $p \in \mathbb{R}^n$ ; does there exist $x \in \Omega$ such that $f(x)=p$ ?

When $f$ is linear, a sufficient condition so that the equation $f(x)=p$ have a solution is $\det(f) \neq 0$ . In the same way, the degree $\deg(f,\Omega,p)$ of $f$ at $p$ with respect to $\Omega$ will be an integer such that $\deg(f,\Omega,p) \neq 0$ implies the existence of a solution in $\Omega$ to the equation $f(x)=p$ .

Brouwer’s Topological Degree (I): Construction and first properties

The key observation is that the quantity $\sum\limits_{x \in f^{-1}(p)} \mathrm{sign}(J_f(x))$ , where $J_f$ is the jacobian determinant of $f$ , is pretty much easier computable than $\mathrm{card} f^{-1}(p)$ , and turns out not to be a differential property but a topological one, in the sense that it is (almost) invariant by homotopy. Of course, we may notice that if the quantity is non-zero, then the equation $f(x)=p$ has a solution (although the converse is false).

The challenge is then to extend this expression for continuous maps.

In this first note, we give a possible construction using an integral interpretation of the previous expression, more handable for our purpose.

Brouwer’s Topological Degree (II): An Axiomatic Definition

However, our definition is not really useful for concrete computations of degrees. In this second note, we identify three properties that caracterize completely Brouwer’s degree: normality, additivity and homotopy invariance.

In fact, it will be the only properties that will be really useful for applications together with the expression given above for regular functions.

Brouwer’s Topological Degree (III): First Applications

In our third note, we present some applications of Brouwer’s degree in real analysis, topology and complex analysis. Of course, many of them are related to the resolution of equations of the type $f(x)=p$ , solving problems about surjectivity, fixed points or zeroes.

Brouwer’s Topological Degree (IV): Jordan Curve Theorem

In this note we present a surprising and nontrivial application of Brouwer’s degree by proving Jordan curve theorem.

Brouwer’s Topological Degree (V): Borsuk Theorem

We conclude our series with a second nontrivial application by proving Borsuk theorem. Also, as a corollary, the invariance of domain theorem will be proved.

As references, we mainly used:

Erhard Heinz, An Elementary Analytic Theory of the Degree of Mapping in n-Dimensional Space, Indiana Univ. Math. J. 8 No. 2 (1959), 231–247.
Donal O’Regan, Yeal Je Cho, Yu-Qing Chen, Topological Degree Theory and Applications, Chapman and Hall/CRC.
Jérôme Droniou, Degrés Topologiques et Applications (in French).

Topology

Brouwer's degree, Topological degree

August 12, 2013

2 Comments

John

Very nice introduction to this fancy subject! But I have one question: Are there simple examples that solution of equation f(x)=p does not always imply that degree is non-zero? Case R^1 is of course trivial as a Bolzano’s theorem.

Permalink, Reply

January 17, 2015 9:25 am

chiasme

A lot of examples can be constructed in the following way:

Let $f_i : \Omega_i \to \Omega_i$ be $n$ smooth functions and define $f : \Omega_1 \times \cdots \times \Omega_n \to \Omega_1 \times \cdots \times \Omega_n$ by $(x_1, \ldots, x_n) \mapsto (f_1(x_1), \ldots , f_n(x_n) )$ . Then, $J_f(x_1, \ldots, x_n) = \prod\limits_{i=1}^n f_i' (x_i)$ so that $\mathrm{deg} (f)= \sum\limits_{x \in f^{-1}(p) } \prod\limits_{i=1}^n f_i' (x_i)$ for some regular value $p$ .

For instance, let $f : (x,y) \mapsto (\sin(x), \sin(y))$ and consider the equation $f(x,y)= (0,0)$ . Of course, there are a lot of solutions, more precisely $f^{-1}(0,0) = \{ (p \pi, q \pi) \mid p,q \in \mathbb{Z} \}$ . But, considering the domain $\Omega = ( - \pi- 1, 2 \pi +1) \times (- \pi-1, \pi +1)$ , and because $J_f(x,y)= \cos (x) \cdot \cos(y)$ , it is easy to compute that $\mathrm{deg} (f)= 0$ .