Let $f,g:X\rightarrow S^n$ be maps satisfying $f(x) + g(x) \not= 0$ for all $x\in X$. Prove $f\simeq g$.

Question

I was assigned this problem for homework. I am not exactly sure where to start with this proof. Any suggestions?

Answer 1

Remark that for $t\in [0,1]$, $tf(x)+(1-t)g(x)=0$ implies that $tf(x)=(1-t)g(x)=0$, since $\|f(x)\|=\|g(x)\|=1$, we deduce that $t=1-t$ and $t=1/2$ and $f(x)+g(x)=0$ impossible, so you can define Write $H(t,x)={{tf(x)+(1-t)g(x)}\over{\|tf(x)+(1-t)g(x)\|}}$