MHB What is the Homological Degree of a Fixed Point Free Continuous Map?

  • Thread starter Thread starter Euge
  • Start date Start date
  • Tags Tags
    2016
Euge
Gold Member
MHB
POTW Director
Messages
2,072
Reaction score
245
Here is this week's POTW:

-----
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 $(-1)^{n+1}$.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
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, $\deg(f) = \deg(-\mathbf 1) = (-1)^{n+1}$.
 
Back
Top