Prove that the Brouwer theorem is false for the open ball $|x|^2

Question

Prove that the Brouwer theorem is false for the open ball $|x|^2 <a$

Brouwer Theorem: Any smooth map $f$ of the close unit ball $B^n \subset R^n$ tin to it self must have a fixed point.

I need to find a counter example, so I let $f: B^k \to B^k$ be a composition function

$$B^k \to^g R^k \to^h R^k \to^{g^{-1}} B^k$$

where $h$ be define as $h(x) =x+a$ for $a \not =0$ . Note that $h: R^k \to R^k$ has no fixed point, so $f: B^k \to B^k$ doesn't have any fixed point either.

Is my reasoning acceptable? or it need more work? If so, please help me improve it.

However, $f(0, \ldots, 0) = (0, \ldots, 0)$, so you need to find a different function. — Sammy Black, Oct 12 '14 at 22:27
Induction would not be useful. Try to think of a map of the closed ball whose only fixed point is on the boundary. Then take its restriction to the open ball. — , Oct 12 '14 at 22:29
This function maps $B^k$ diffeomorphically to all of $\Bbb R^k$. Can you find a function on $\Bbb R^k$ with no fixed point and transport it back to $B^k$? — Ted Shifrin, Oct 13 '14 at 00:48
Looks good, but you deleted the definition of $g$ when you edited! — Ted Shifrin, Oct 14 '14 at 02:59
@TedShifrin you mean this function $g(x)= \frac {ax}{\sqrt {a^2 +|x|^2}}$? — Diane Vanderwaif, Oct 14 '14 at 09:38

score 5 · Answer 1 · answered Jan 18 '15 at 02:41

As has been established in the comments, your construction works because $f$ has a fixed point if and only if $h$ does, but $h$ was chosen so that it doesn't have a fixed point.

Another map that works is $f : B(0, a) \to B(0, a)$ given by $f(x) = \frac{1}{2}(x+ae_1)$ where $e_1 = (1, 0, \dots, 0)$. This sends a point $x \in B(0, a)$ to the midpoint of the line segment between $x$ and $ae_1 \in \partial B(0, a)$. It can be verified either algebraically or geometrically that $f$ has no fixed point. If the map were extended to $\overline{B(0, a)}$, it would have a fixed point at $ae_1$.

Prove that the Brouwer theorem is false for the open ball $|x|^2

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