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I want to compute the fundamental group of $\{x : \|x\| < 1\}$. It should be trivial as I can contract to one point. What is wrong with the following method?

Consider the map $H(x, t) = (1 - t)x + tx/(2\|x\|)$. We claim that $H$ is a deformation retraction of $X = \{x : \|x\| < 1\}$ onto $A = \{x : \|x\| = 1/2\}$. We have $H(x, 0) = x$ and $H(x, 1) \in A$ for all $x \in X$ and $H(a, t) = a$ for all $a \in A$. Since $A$ is homeomorphic to $S^{1}$, it follows that $\pi_{1}(X) = \mathbb{Z}$. What is wrong with this proof?

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3 Answers 3

Your retraction is not defined at $x=(0,0)$. If you had considered the punctured disk there would be no problem.

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Your map is ill-defined, e.g. since you have

$$ H(\vec{0},1) = \vec{0} / 0$$

and there is no way to continuously extend $H$ to fill things in.

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the problem is that the map $x \mapsto x/||x||$ is not well defined in $0$.

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