# Topological degree theory

Let $D$ be a region in $R^n$. If $f:D\to R^n$ is continuous, nonzero on $\partial D$ and of Brower degree 0, does there exists a continuous function $g=f$ on $\partial D$ and $g\neq 0$ on $D$?

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The Brower degree is usually defined for maps $f:M\to N$ that send $\partial M$ into $\partial N$ and does not seem to extend to this setting. What is the degree of the map $f:[-1,1] \to \mathbb R$ which sends linearly $[-1,0]$ to $[-1,0]$ and $[0,1]$ to $[0,-1/2]$? – Bruno Martelli Nov 24 '10 at 11:46
Is the region an open subset with smooth boundary? Is it open? Is it closed? Is its boundary smooth? Does it have non-empty interior? – André Henriques Nov 24 '10 at 19:58
Bruno, I think the OP means degree w.r.to the point $p=0$: the setting is the Brouwer degree attached to a triple $(D,f,p),$ where $D$ is a bounded open subset of $\mathbb{R}^n;$ $f:\bar D\to \mathbb{R}^n$ and $p\in\mathbb{R}^n\setminus f(\partial D)$. – Pietro Majer Nov 24 '10 at 23:15

If $D$ is a smooth compact manifold with boundary, then any $f: \partial D \to \mathbb{R}^n \setminus 0$ of degree $0$ can be extended to a map $D \to \mathbb{R}^n \setminus 0$. Proof: A theorem of Hopf asserts that homotopy classes of maps $\partial D \to S^{n-1}$ are determined by their degree. Thus your map $f|_{\partial D}$ is nullhomotopic. Use the nullhomotopy to extend $f$ over a small interior collar of $\partial D$ inside $D$. On the interior boundary of the collar, it is constant and can be extended by the constant map to all of $D$.
Yes, thank you. It still seems not to be completely clear to me what happens if $\partial D$ is not smooth; then there might not be a neighborhood of $\partial D$ in $D$ homeomorfic to $\partial D\times [0,1]$, but this can be probably solved. – Peter Franek Nov 24 '10 at 22:49
I assume that the setting is as described by Pietro in the comments above, so $D$ is a connected open set and $f(\partial D)$ does not contain $0$. You can perturb $f$ so that $f|_D$ is smooth and $0$ is a regular value. Then $f^{-1}(0)$ consists of finitely many points $x_1,\ldots, x_{2n}$ contained in the open domain $D$ and $f$ is a local diffeomorphism at each $x_i$. Since the degree is zero, half of these local diffeomorphisms are orientation-preserving, say on $x_1,\ldots, x_n$. The others are orientation-reversing. Choose $n$ disjoint smooth arcs in $D$ that connect $x_i$ to $x_{i+n}$. Take a small regular neighborhood of each arc: it is an open ball with smooth boundary. On each such ball $B$ the map $f|_B$ has degree zero and you can apply Ebert's argument to modify $f|_B$ such that it avoids $0$.