Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

By the degree argument we see that any reflection on $\ S^{2n}$ is homotopic to the antipodal map.

But that seems a big theorem. I am looking for a straightforward argument.

share|improve this question
    
Hint: Consider a suitable path in $\mathrm{O}(2 n + 1)$ from $\textrm{diag}(1, \ldots, 1, -1)$ to $\textrm{diag}(-1, \ldots, -1, -1)$. –  Zhen Lin Dec 9 '12 at 17:59
    
I think you mean a reflection specifically in a hyperplane, right? For example, reflecting $S^2$ in any axis in $\mathbb{R}^3$ is a half-turn about that axis, which is homotopic to the identity map. –  Hew Wolff Dec 9 '12 at 22:46

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.