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<j \\ \partial_{l+1} F & \text{if} \ l \geq j \end{array} \right..

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

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<j-1} a_{jl} + \sum\limits_{l<j} a_{jl}=0. \hspace{1cm} \square

Advertisements