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Problem of the Week #234 - Nov 22, 2016

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Euge

MHB Global Moderator
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Jun 20, 2014
1,896
Here is this week's POTW:

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Let $n$ be a positive integer, and let $\Bbb S^n \to \Bbb S^n$ be a fixed point free continuous map. Show that the map's homological degree is $(-1)^{n+1}$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,896
No one answered this week's problem. You can read my solution below.


Since $f(x) \neq x$ for all $x\in \Bbb S^n$, there is a homotopy from $f$ to the antipodal map $-\bf 1$ given by $h_t(x) = \frac{(1 - t)f(x) - tx}{\|(1 - t)f(x) - tx\|}$, for all $t\in [0,1]$ and $x\in \Bbb S^n$. Thus, $\deg(f) = \deg(-\mathbf 1) = (-1)^{n+1}$.
 
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