Borsuk Ulam Theorem - nullhomotopic

I am going through the proof of Borsul Ulam theorem in the book Algebraic Topology by Hatcher. Here is the snippet.  What I am unable to understand is the line in the proof where it said that "since $q$ is odd", $h$ is not nullhomotopic. Say in $\pi(S^1)$, we have $( cos(2\pi s), sin(2\pi s))$ as the generator element. Then if $q$ is odd, then as per the claim in the proof, $[ cos(6\pi s), sin(6\pi s)]=h (s)$ assuming $q=3$ is not nullhomotopic. But we can define a nullhomotopy on this circle as defined below ($x_0$ is the base point):

$$f_t(s) = (1-t)h(s) + x_0t$$

So I can't understand on why if $q$ is odd, $h$ is not nullhomotopic , because as described above, I can define a nullhomotopy on this circle.

Also why $h$ is NOT nullhomotopic when $q$ is odd, but nullhomotopic when $q$ is even.

What I am getting wrong here. Support would be greatly appreciated.

Your map $f_t(s)$ does not take values on the circle: it is a nullhomotopy for maps from $S^1$ to $\Bbb R^2$, not a nullhomotopy (as requires) for maps from $S^1$ to $S^1$.
• Can you please give an example of a nullhomotopy when $q$ is even? I think in case $q$ is even, the condition: $h(s+1/2)=-h(s)$ won't be satisfied. – user3001408 Jul 1 '17 at 11:33
• $\pi^1(S^1)$ is infinite cyclic. The point is that odd numbers are automatically nonzero. Your element corresponds to $q$ times a generator of $\pi^1(S^1)$ so cannot be nullhomotopic. – Lord Shark the Unknown Jul 1 '17 at 15:30