the nonexistence of a division algebra on $\mathbb R^{2n+1}$

Question

Prove that there is no division algebra structure on $\mathbb R^{2n+1}$.

The hint says

Suppose that there is such a structure on $\mathbb R^{2n+1}$. Take a nonzero $a \in \mathbb R^{2n+1}$. Consider $f: S^{2n} \rightarrow S^{2n}, x \mapsto \frac{ax}{|ax|}$. Prove that $f$ and $-f$ are homotopic.

But how is this homotopy constructed? Thanks for the help.

Thomas Andrews · Accepted Answer · 2013-05-30T17:31:35.747

2

Hint: Find a path between $a$ and $-a$ that does not go through $0$.

edited May 30 '13 at 17:31

answered May 30 '13 at 16:40

Thomas Andrews

186,215

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Of course $n=0$ is excluded here. – GEdgar May 30 '13 at 17:35
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Which is good, since $\mathbb R^1$ can be made into a division algebra pretty easily :) @GEdgar – Thomas Andrews May 30 '13 at 17:50
Thank you very much for the answer. I think I know what to do now. – sunkist Jun 17 '13 at 12:53

the nonexistence of a division algebra on $\mathbb R^{2n+1}$

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