Here is the first note of a series of posts devoted to Brouwer’s topological degree. Our aim is to give a construction of Brouwer’s topological degree.

Lemma 1: Let $\Omega \subset \mathbb{R}^n$ be a bounded open subspace, $y : \overline{\Omega} \to \mathbb{R}^n$ be a continuous map $C^2$ on $\Omega$, $p \notin y(\partial \Omega)$ and $\Phi : \mathbb{R} \to \mathbb{R}$ be a function satisfying:

• $\Phi$ is continuous on $[0,+ \infty)$ and vanishes on a neighborhood of $0$ and on $[\epsilon,+ \infty)$ for some $\epsilon< d(p,y(\partial \Omega))$,
• $\displaystyle \int_{\mathbb{R}^n} \Phi( \|x\| )dx=1$.

Then $\displaystyle \int_{\Omega} \Phi(\|y(x)-p\|) J_y(x)dx$ does not depend on $\Phi$, where $J_y$ is the jacobian determinant of $y$.

Lemma 2: Let $\Omega \subset \mathbb{R}^n$ be a bounded open subspace, $y : \overline{\Omega} \to \mathbb{R}^n$ be a continuous map $C^2$ on $\Omega$ and $\varphi : [0,+ \infty) \to \mathbb{R}$ be a continuous function. Suppose that:

• $\| y(x) \| > \epsilon >0$ for all $x \in \partial \Omega$,
• $\varphi$ vanishes on a neighborhood of $0$ and on $[\epsilon,+ \infty)$,
• $\displaystyle \int_0^{\infty} r^{n-1} \varphi(r)dr=0$.

Then $\displaystyle \int_{\Omega} \varphi( \| y(x) \|) J_y(x)dx=0$.

Proof. Let $\psi : r \mapsto \left\{ \begin{array}{ccc} r^{-n} \int_0^r \rho^{n-1} \varphi(\rho) d \rho & \text{if} \ r >0 \\ 0 & \text{otherwise} \end{array} \right.$ and $f^j : x \mapsto \psi (\|x\|) x_j$ for $1 \leq j \leq n$. Let $A_{ij}$ denote the $(i,j)$-cofactor of the jacobian matrix of $y$.

First, notice that $\psi$ vanishes on a neighborhood of $0$ and $f^j \circ y$ on a neighborhood of $\partial \Omega$ since $\psi(r)= r^{-n} \int_0^{+ \infty} \rho^{n-1} \varphi(\rho) d \rho=0$ when $r> \epsilon$. Therefore, $\psi$ is $C^1$ and $x \mapsto A_{ij}(x)f^j(y(x))$ can be extended on $\mathbb{R}^n$ by zero. Then

$\begin{array}{ll} \Delta(x) & \displaystyle = \sum\limits_{i=1}^n \frac{\partial}{\partial x_i} \sum\limits_{j=1}^n A_{ij}(x) f^j(y(x)) \\ \\ & \displaystyle = \sum\limits_{j=1}^n \underset{=0}{\underbrace{ \left( \sum\limits_{i=1}^n \frac{\partial A_{ij}}{\partial x_i} (x) \right)}} f^j(y(x)) + \sum\limits_{j=1}^n \sum\limits_{k=1}^n \underset{= \delta_{k,j}J_y(x)}{\underbrace{ \left( \sum\limits_{i=1}^n A_{ij}(x) \frac{\partial y_k}{\partial x_i}(x) \right)}} \frac{\partial f^j}{\partial x_k} (y(x)) \\ \\ & \displaystyle = J_y(x) \sum\limits_{j=1}^n \frac{\partial f^j}{\partial x_j} (y(x)) = J_y(x)(n \psi( \|y(x) \|)+ \|y(x) \| \psi'(\|y(x)\|)) \\ \\ & \displaystyle = J_y(x) \varphi (\|y(x)\|) \end{array}$

We deduce that

$\begin{array}{ll} \displaystyle \int_{\Omega} \varphi(\|y(x)\|) J_y(x)dx & \displaystyle = \sum\limits_{i,j=1}^n \int_{\Omega} \frac{\partial}{\partial x_i} (A_{ij}(x) f^j(y(x))) dx \\ \\ & \displaystyle = \sum\limits_{i,j=1}^n \int_{\mathbb{R}^n} \frac{\partial}{\partial x_i} (A_{ij}(x) f^j(y(x))) dx \\ \\ & \displaystyle = \sum\limits_{i,j=1}^n \int_{\mathbb{R}^{n-1}} \left( \int_{\mathbb{R}} \frac{\partial}{\partial x_i} (A_{ij}(x) f^j(y(x)))dx_i \right) dx_1 \dots dx_{i-1}dx_{i+1} \dots dx_n \\ \\ & =0 \hspace{1cm} \square \end{array}$

Notice that we used a property on cofactors of a jacobian matrix whose a proof will be found at the end of this note for convenience:

Lemma 3: Let $f: \mathbb{C}^n \to \mathbb{R}^n$ be a $C^2$-map and let $\Delta_{i,j}(x)$ denote the $(i,j)$-cofactor of the facobian matrix of $f$. Then for all $1 \leq i,j \leq n$ and $x \in \mathbb{R}^n$,

$\displaystyle \sum\limits_{j=1}^n \partial_j \Delta_{i,j}(x)=0$ and $\displaystyle \sum\limits_{j=1}^n \partial_j f_k(x) \Delta_{i,j}(x)= \delta_{i,k} J_f(x)$.

Proof of lemma 1. Let $D$ be the vector space of applications $\mathbb{R} \to \mathbb{R}$ satisfying the first point of our lemma and consider the following linear functional:

$\displaystyle L \Phi= \int_0^{+ \infty} r^{n-1} \Phi(r)dr, \ M \Phi = \int_{\mathbb{R}^n} \Phi(\|x\|)dx, \ N \Phi = \int_{\Omega} \Phi(\|y(x)-p\|)J_y(x)dx.$

Applying lemma 2 to $y-p$ and $y$, we deduce that $L \Phi=0$ implies $M \Phi=N \Phi=0$.

Let $\Phi_1, \Phi_2 \in D$ satisfying $M\Phi_1=M \Phi_2=1$. Because $L(L \Phi_1 \cdot \Phi_2- L \Phi_2 \cdot \Phi_1)=0$, we deduce that $0= M(L \Phi_1 \cdot \Phi_2- L \Phi_2 \cdot \Phi_1)= L(\Phi_1- \Phi_2)$ hence $N(\Phi_1- \Phi_2)=0$ ie. $N \Phi_1=N \Phi_2$. $\square$

Now we are able to define Brouwer’s degree of a $C^2$ map.

Definition: Let $\Omega \subset \mathbb{R}^n$ be a bounded open subspace, $f : \overline{\Omega} \to \mathbb{R}^n$ be a continuous map $C^2$ on $\Omega$ and $p \notin f( \partial \Omega)$. With the previous notations, we define the degree of $f$ at $p$ with respect to $\Omega$ by

$\displaystyle \deg(f,\Omega,p) = \int_{\Omega} \Phi(\|f(x)-p\|) J_f(x)dx$.

In fact, it is possible to simplify the definition when $p$ is a regular value of $f$.

Lemma 4: Let $\Omega \subset \mathbb{R}^n$ be a bounded open subspace, $f : \overline{\Omega} \to \mathbb{R}^n$ be a continuous map $C^2$ on $\Omega$ and $p \notin f(\partial \Omega)$ be a regular value of $f$. Then there exists $\epsilon_0>0$ such that for all continuous map $\phi : \mathbb{R}^n \to \mathbb{R}$ satisfying

• $\displaystyle \int_{\mathbb{R}^n} \phi(x)dx=1$,
• $\phi(x)=0$ while $\|x\| \geq \epsilon_0$,

we have $\displaystyle \deg(f,\Omega,p)= \int_{\Omega} \phi(f(x)-p)J_f(x)dx$.

Proof. Let $f^{-1}(p) = \{x_1, \dots, x_m\}$. There exists $\delta>0$ such that $f$ defines a diffeomorphism from $B(x_i,\delta)$ and $B(x_i , \delta) \cap B(x_j,\delta)= \emptyset$ for $i \neq j$. Let $\eta>0$ such that $B(p,\eta) \subset \bigcap\limits_{i=1}^m f(B(x_i,\delta))$ et let $U_i=f^{-1}(B(p,\eta)) \cap B(x_i,\delta)$.

We show that any $\epsilon_0<\eta$ works. Let $\phi : \mathbb{R}^n \to \mathbb{R}$ be a function satisfying the two points of our lemma.

Notice that if $x \notin \bigcup\limits_{i=1}^m U_i= f^{-1}(B(p,\eta))$ then $\|f(x)-p\| \geq \eta > \epsilon_0$ so $\phi(f(x)-p)=0$. Thus

$\displaystyle \int_{\Omega} \phi(f(x)-p)J_f(x)dx= \sum\limits_{i=1}^m \int_{U_i} \phi(f(x)-p)J_f(x)dx.$

Because $J_f(x)$ has the same sign for all $x \in U_i$,

$\begin{array}{ll} \displaystyle \int_{\Omega} \phi(f(x)-p) J_f(x)dx & \displaystyle = \sum\limits_{i=1}^m \int_{U_i} \phi(f(x)-p)J_f(x)dx \\ \\ & \displaystyle = \sum\limits_{i=1}^m \mathrm{sign}(J_f(x_i)) \int_{U_i} \phi(f(x)-p) \cdot |J_f(x)| dx \\ \\ & \displaystyle = \sum\limits_{i=1}^m \mathrm{sign}(J_f(x_i)) \int_{U_i} \phi(f(x)-p) \cdot J_{f-p}(x)| dx \\ \\ & \displaystyle = \sum\limits_{i=1}^m \mathrm{sign}(J_f(x_i)) \int_{f(U_i)-p} \phi(x)dx \\ \\ & \displaystyle = \sum\limits_{i=1}^m \mathrm{sign}(J_f(x_i)) \int_{B(0,\eta)} \phi(x)dx \\ \\ & \displaystyle = \sum\limits_{i=1}^m \mathrm{sign}(J_f(x_i)) \int_{\mathbb{R}^n} \phi(x)dx = \sum\limits_{i=1}^m \mathrm{sign}(J_f(x_i)). \ \square \end{array}$

Corollary 1: Let $\Omega \subset \mathbb{R}^n$ be a bounded open subspace, $f : \overline{\Omega} \to \mathbb{R}^n$ be a continuous map $C^2$ on $\Omega$ and $p \notin f(\partial \Omega)$ be a regular value of $f$. Then

$\displaystyle \deg(f,\Omega,p)= \sum\limits_{x \in f^{-1}(p)} \mathrm{sign} (J_f(x))$.

Now we want to extend our definition for continuous functions. The key observation is that in fact $\deg$ is not a differential property:

Lemma 5: Let $\Omega \subset \mathbb{R}^n$ be a bounded open subspace, $y_1,y_2 : \overline{\Omega} \to \mathbb{R}^n$ be two continuous maps $C^2$ on $\Omega$, $p \in \mathbb{R}^n$ and $\epsilon>0$ such that:

• $\|y_i(x)-p \| >7 \epsilon$ for all $x \in \partial \Omega$,
• $\|y_1-y_2\|_{\infty} < \epsilon$

Then $\deg(y_1,\Omega,p)= \deg(y_2,\Omega,p)$.

Proof. Because $\deg(y_i,\Omega,p)= \deg(y_i-p,\Omega,0)$, we may suppose without loss of generality that $p=0$.

Let $f : \mathbb{R} \to [0,1]$ be a $C^1$-map satisfying $f(r)= \left\{ \begin{array}{cl} 1 & \text{if} \ 0 \leq r \leq 2 \epsilon \\ 0 & \text{if} \ 2 \epsilon \leq r \leq 3 \epsilon \end{array} \right.$ and let $y_3 : x \mapsto (1-f(\|y_1(x)\|))y_1(x) + f(\|y_1(x)\|)y_2(x)$.

In particular, $y_3(x)= \left\{ \begin{array}{cl} y_1(x) & \text{if} \ \|y_1(x)\| \geq 3 \epsilon \\ y_2(x) & \text{if} \ \|y_1(x)\| \leq 2 \epsilon \end{array} \right.$.

Let $\Phi_1,\Phi_2 : [0,+ \infty) \to \mathbb{R}$ be two continuous maps vanishing on a neighborhood of $0$ and satisfying

$\displaystyle \int_{\mathbb{R}^n} \Phi_1(\|x\|)dx= \int_{\mathbb{R}^n} \Phi_2(\|x\|)dx=1$ and $\left\{ \begin{array}{cl} \Phi_1(x)=0 & \text{if} \ r \leq 4 \epsilon \ \text{or} \ r \geq 5 \epsilon \\ \Phi_2(r)=0 & \text{if} \ r \geq \epsilon \end{array} \right.$.

Notice that

$\Phi_2(\|y_3(x)\|) J_{y_3}(x)= \left\{ \begin{array}{cl} \Phi_2(\|y_2(x)\|) J_{y_2}(x) & \text{if} \ \|y_1(x) \| \leq 2 \epsilon \\ 0 & \text{if} \ \|y_1(x)\| > 2\epsilon \end{array} \right.$

and

$\Phi_1(\|y_3(x)\|) J_{y_3}(x) = \left\{ \begin{array}{cl} \Phi_1(\|y_1(x)\|) J_{y_1}(x) & \text{if} \ \|y_1(x)\| \geq 3 \epsilon \\ 0 & \text{if} \ \|y_1(x)\| < 3 \epsilon \end{array} \right.$

Moreover, if $\|y_1(x)\|< 3 \epsilon$ then $\Phi_1(\|y_1(x)\|)=0$; if $\|y_(x)\|> 2 \epsilon$ then $\|y_2(x)\| \geq \|y_1(x) \| - \|y_1-y_2\|_{\infty} >2 \epsilon-\epsilon=\epsilon$ hence $\Phi_2(\|y_2(x)\|)=0$.

Thus $\left\{ \begin{array}{l} \Phi_2(\|y_3(x)\|) J_{y_3}(x) = \Phi_2(\|y_2(x)\|) J_{y_2}(x) \\ \Phi_1(\|y_3(x)\|) J_{y_3}(x) = \Phi_1(\|y_1(x)\|) J_{y_1}(x) \end{array} \right.$.

Because $7 \epsilon < d(p,y_i(\partial \Omega))$ and

$\begin{array}{ll} \|y_3(x)-p \| & = \| y_1(x)-p+f(\|y_1(x) \|) (y_2(x)-y_1(x)) \| \\ & \geq \|y_1(x)-p \| - \|y_2-y_1\|_{\infty} \\ & \geq 7 \epsilon - \epsilon = 6 \epsilon \end{array}$

for all $x \in \partial \Omega$, we deduce by integrating the equalities above that $\deg(y_1,\Omega,0)= \deg(y_2,\Omega,0)$. $\square$

Now we are ready to introduce our definition (we implicitely use Stone-Weierstrass approximation theorem):

Definition: Let $\Omega \subset \mathbb{R}^n$ be a bounded open subspace, $f : \overline{\Omega} \to \mathbb{R}^n$ be a continuous map and $p \notin f(\partial \Omega)$. We define

$\deg(f,\Omega,p)= \lim\limits_{n \to + \infty} \deg(f_n,\Omega,p)$

where $(f_n)$ is any sequence of continuous maps $\overline{\Omega} \to \mathbb{R}^n$ $C^2$ on $\Omega$ converging uniformely to $f$ and satisfying $p \notin f_n(\partial \Omega)$ for all $n \geq 0$.

The main properties of Brouwer’s degree are summarized in the following property:

Property 6: Brouwer’s degree $\deg$ satisfies:

• (normality) $\deg(\mathrm{Id},\Omega,p)=1$ if $p \in \Omega$,
• (additivity) Let $\Omega_1,\Omega_2 \subset \Omega$ be two disjoint bounded open subspaces such that $p \notin f( \overline{\Omega} \backslash ( \Omega_1 \cup \Omega_2 ))$. Then

$\deg(f,\Omega,p)= \deg(f,\Omega_1,p)+ \deg(f,\Omega_2,)$.

• (homotopy invariance) Let $H : [0,1] \times \overline{\Omega} \to \mathbb{R}^n$ be a homotopy and $y : [0,1] \to \mathbb{R}^n$ be a continuous map satisfying $y(t) \notin H(t,\partial \Omega)$ for all $t \in [0,1]$. Then

$\deg(H(0, \cdot),\Omega,y(0))= \deg(H(1, \cdot),\Omega,y(1))$.

Proof. Normality and additivity are consequences of the definition and corollary 1.

Let $H : [0,1] \times \overline{\Omega} \to \mathbb{R}^n$ be a continuous map and $p\notin H(t, \partial \Omega)$ for all $t \in [0,1]$.

Because $[0,1] \times \partial \Omega$ is compact, there exists $\epsilon>0$ such that $\|H(t,x)-p\| >7 \epsilon$ for all $t \in [0,1]$ and $x \in \partial \Omega$; because $[0,1] \times \overline{\Omega}$ is compact, $H$ is uniformely continuous so there exists $\delta >0$ such that for all $t_1,t_2 \in [0,1]$, $|t_1-t_2|< \delta$ implies $|H(t_1,x)-H(t_2,x)|< \epsilon$ for all $x \in \overline{\Omega}$.

Let $0=t_0 < t_1 < \dots < t_{m-1}< t_m=1$ be a partition of $[0,1]$ satisfying $|t_{i+1}-t_i|<\delta$ for all $0 \leq i \leq m-1$. For all $0 \leq i \leq m$, let $(f_{i,k} : \overline{\Omega}\to \mathbb{R}^n)$ be a sequence of continuous maps $C^2$ on $\Omega$ converging uniformely to $H(t_i, \cdot)$.

Notice that for all $x \in \partial \Omega$ and for $k$ large enough,

$\|f_{i,k}(x)-p \| > \| H(t_i,x)-p\| - \| f_{i,k}(x)-H(t_i,x) \| > 7 \epsilon$,

and for all $x \in \Omega$ and for $k$ large enough,

$\|f_{i+1,k}(x)-f_{i,k}(x)\| < \| f_{i+1,k}(x)-H(t_{i+1},x) \| + \|f_{i,k}(x)-H(t_i,x) \| + \| H(t_{i+1},x)-H(t_i,x) \| < \epsilon$.

According to lemma 5, we deduce that $\lim\limits_{k \to + \infty} \deg(f_{i,k},\Omega,p)= \lim\limits_{k \to + \infty} \deg(f_{i+1,k}, \Omega,p)$. Thus

$\deg(H(0, \cdot) , \Omega,p) = \deg (H(t_0, \cdot), \Omega,p) = \deg (H(t_1, \cdot), \Omega,p) = \deg (H(t_2, \cdot),\Omega,p) = \dots = \deg( H(t_m, \cdot),\Omega ,p) = \deg (H(1, \cdot), \Omega,p)$.

To conclude the proof, it is sufficient to prove that $\deg(f, \Omega, \cdot)$ is constant on any connected component of $\mathbb{R}^n \backslash f(\partial \Omega)$. For that, we show that $\deg(f,\Omega, \cdot)$ is locally constant.

Let $\epsilon>0$. Let $z_1,z_2 \in \mathbb{R}^n$ be two points satisfying $d(z_i,f(\partial \Omega))> 7 \epsilon$ and $\|z_1-z_2\| < \epsilon$. Then $\deg(f,\Omega,z_i)= \deg(f-z_i,\Omega,0)$ and we conclude thanks to lemma 5. $\square$

Proof of lemma 3. Let $J$ be the jacobian matrix of $f$. It is known that $^t \mathrm{com}(A) \cdot A = \det(A) \cdot \mathrm{I}_n$ for any matrix $A \in M_n(\mathbb{R})$. Now notice that

$\left( ^t\mathrm{com}(J(x)) J(x) \right)_{ik}= \sum\limits_{j=1}^n \Delta_{ij}(x) \partial_j f_k(x)$ and $(J_f(x) \mathrm{I}_n )_{ik}= \delta_{ik} J_f(x)$.

Then the second equality of our lemma follows easily.

For the first equality, let $1 \leq i \leq n$ be fixed and let

$F= \ ^t (f_1,\dots,f_{i-1},f_{i+1}, \dots, f_n)$ and $X_{jl} = \left\{ \begin{array}{cl} \partial_l F & \text{if} \ l.

Thus, $\Delta_{ij}(x)= (-1)^{i+j} \det(X_{j1},\dots,X_{j,n-1})$. Therefore

$\displaystyle (-1)^i \partial_j \Delta_{ij}(x)= \sum\limits_{l=1}^{n-1} \underset{ a_{jl} }{ \underbrace{ (-1)^j \det (X_{j1}, \dots, X_{j,l-1}, \partial_j X_{jl}, X_{j,l+1}, \dots, X_{j,n-1}) }},$

hence

$\displaystyle (-1)^i \sum\limits_{j=1}^n \partial_j \Delta_{ij}(x) = \sum\limits_{j=1}^n \sum\limits_{l=1}^{n-1} a_{jl} = \sum\limits_{(j,l) \in E} a_{jl} + \sum\limits_{(j,l) \in F} a_{jl}$

where $E= \{(j,l) \mid 1 \leq l < j \leq n \}$ and $F= \{(j,l) \mid 1 \leq j \leq l \leq n-1\}$. Noticing that $(a,b) \mapsto (b,a-1)$ defines a bijection from $E$ to $F$, we finally get

$\displaystyle \sum\limits_{j=1}^n \partial_j \Delta_{ij}(x) = (-1)^i \sum\limits_{(j,l) \in E} (a_{jl}+a_{l,j-1}).$

Let $(j,l) \in E$. Notice that

• If $b then $X_{lb}=\partial_b F=X_{jb}$,
• If $l \leq b then $X_{lb}= \partial_{b+1} F= X_{j,b+1}$,
• If $b \geq j >l$ then $X_{lb}= \partial_{b+1} F=X_{jb}$,
• Because $l and using Schwarz’ lemma, $\partial_l X_{l,j-1}= \partial_{lj} F= \partial_{jl} F= \partial_j X_{jl}$.

We deduce the general formula:

$\begin{array}{ll} a_{l,j-1} & = (-1)^l \det ( X_{l1}, \dots, X_{l,j-2}, \partial_l X_{l,j-1}, X_{lj}, \dots, X_{l,n-1} \\ \\ & = (-1)^l \det( X_{j1}, \dots, X_{j,l-1}, X_{j,l+1} , \dots, X_{j,j-1}, \partial_j X_{jl} , X_{jj} , \dots , X_{j,n-1}) \end{array}$

Thus, if $l=j-1$,

$a_{j-1,j-1} = (-1)^{j-1} \det(X_{j1}, \dots, X_{j,j-1}, \partial_j X_{jl}, X_{jj}, \dots, X_{j,n-1}) = -a_{j,j-1}$

and if $l< j-1$,

$\begin{array}{ll} a_{l,j-1} & = (-1)^l \det (X_{j1}, \dots, X_{j,l-1}, \underbrace{ X_{j,l+1}, \dots, X_{j,j-1} }, \partial_j X_{jl}, X_{jj}, \dots, X_{j,n-1}) \\ \\ & = (-1)^l (-1)^{j-1-l} \det ( X_{j1}, \dots, X_{j,l-1}, \partial_j X_{jl}, \underbrace{ X_{j,l+1}, \dots, X_{j,j-1} } , X_{jj}, \dots, X_{j,n-1}) \\ \\ & = -a_{jl} \end{array}$

Therefore,

$\displaystyle \sum\limits_{(j,l) \in E} (a_{jl}+ a_{l,j-1}) = -a_{j,j-1}- \sum\limits_{l