Show that $f$ is homotopic to $g$.

Question

Let $X$ be any topological space. If $f,g:X\to S^n$ (n-sphere) are continuous, such that $f(x)$ and $g(x)$ are never antipodal, show that $f$ is homotopic to $g$.

I have no idea about this, please guve me some direction. Thanks.

Answer 1

The basic idea is the the line joining $f(x)$ and $g(x)$ never containts the origin and hence can be projected onto the $S^n$. To be more concrete: Define $$ h(t,x) := (1-t)f(x) + tg(x) \in \mathbf R^{n+1} $$ and $$ H(t,x) := \frac{h(t,x)}{|h(t,x)|} $$ Then $H$ is a homotopy from $f$ to $g$.

Answer 2

Consider $F : X\times I \to S^n$ as $F(x,t)= \frac{tf(x)+ (1-t)g(x)}{\left \| tf(x)+ (1-t)g(x) \right \|}$..

$F$ is well defined and gives you the required homotopy.