# Brouwer degree and homotopy invariance

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ and let $f \in C^1(\overline{\Omega})$. Following Topological degree theory and applications, we define the degree of $f$ with respect to some regular point $p \notin f(\partial \Omega)$ by $$\deg(f,\Omega,p) = \sum\limits_{x \in f^{-1}(p)} \text{sign}(J_f(x))$$ where $J_f(x)$ denotes the jacobian determinant of $f$ at $x$. In fact, if $f \in C^2(\overline{\Omega})$, $\deg(f,\Omega,\cdot )$ is locally constant so the degree of $f$ can be defined at a singular point $p$ by taking a regular point $q$ closed to $p$: $$\deg(f,\Omega,p)=\deg(f,\Omega,q), \ \|p-q\| < d(p,f(\partial \Omega))$$ A fundamental property is that if $H \in C^2([0,1] \times \overline{\Omega})$ and $p \notin H_t(\partial \Omega)$ for all $t \in [0,1]$, then $\deg(H_t,\Omega,p)$ does not depend on $t$ (homotopy invariance).

My question is: how to prove it?

• This does not quite address your question, but you might enjoy Erhard Heinz's exposition of degree theory in his classic paper, An Elementary Analytic Theory of the Degree of Mapping in n-Dimensional Space, Indiana Univ. Math. J. 8 No. 2 (1959), 231–247 where a generalization of the result you ask about is proved in theorem 3. Commented Jun 14, 2013 at 22:37
• @Martin: I found my answer in the paper you mention, thank you! Commented Jun 15, 2013 at 7:50

Proposition: Let $\Omega \subset \mathbb{R}^n$ be a bounded open set, $f \in C^1(\overline{\Omega})$ and $p \notin f(\partial \Omega)$ a regular point of $f$. Then there exists $\epsilon_0>0$ such that for any function $\phi : \overline{\Omega} \to \mathbb{R}$ satisfying $\int_{\Omega} \phi=1$ and $\phi(x)=0$ if $\| x \|>\epsilon_0$, $$\deg(f,\Omega,p):=\sum\limits_{x \in f^{-1}(p)} \text{sign}(J_f(x))= \int_{\Omega} \phi(f(x)-p)J_f(x)dx$$
Then, lemma 2 and theorem 3 in An elementary analytic theory of the degree of mapping in n-dimension space can be proved in almost the same way; just use Sard's lemma to justify that $p$ can be supposed regular.
Another important detail: in Heinz's article, it is prove that we can take $\epsilon_0= d(p,f(\partial \Omega))$ whereas we don't know how evolve $\epsilon_0$ with $f$ in the previous proposition. However, such an information is needed to prove lemma 2. So we have to add some details about this point in the previous proof.