Degree of map from $S^n$ to $S^n$ This is an excercise in the book Algebraic Topology - Greenberg and Harper 
Excercise : Let $f$ and $g$ be a map from $S^n$ to $S^n$ such that $f(x)\neq g(x)$ for all $x$ 
Then $f $ is homotopic to $ag$ where $a$ is antipodal map, hence deg($f)=(-1)^{n+1} $deg($g$). 
Here, deg($f$) is a map from $H_n(S^n; {\bf Z})$ to $H_n(S^n; {\bf Z})$ induced from $f$
Question : If $n=2$, and if $f$ and $g$ are rotations with different axes, then $f(x)\neq g(x)$ for all $x$. So deg($f)=$deg($g$)$=1$ and deg$(a) =-1$. So is the exercise right ? 
 A: Your claim (in your question) is wrong.
Take $f:S^2\to S^2$ to be $\pi$-rotation about z-axis, and take $g:S^2\to S^2$ to be $\pi$-rotation about y-axis. Then the point $x=(1,0,0)$ is mapped to its antipode $(-1,0,0)$ via both $f$ and $g$, i.e. moving in two possible directions ("over" and "sideways").
A: I remember having this same misconception when first taking algebraic topology. Your statement about rotations with different axes is the tricky point. In fact, $f$ and $g$ will agree for some $x$. (Actually, for two antipodal points $x$ and $-x.$) As far as I can tell, Euler was the first one to figure this out; he proved that the composition of two rotations of the sphere is a rotation.
As an example, consider the point $P=(1,0,0)$ on the equator of the unit sphere. Consider the rotation $f$ around the north pole by $180^\circ$. This maps $P$ $180^\circ$ along the equator to $f(P)=(-1,0,0)$. Then consider the rotation $g$ around the point $(0,1,0)$ by $180^\circ$. You can check that this also maps $(1,0,0)$ to $(-1,0,0)$. ($180^\circ$ along a line of longitude this time.) You can easily use a similar method to find examples that don't use $180^\circ$ rotations if you find this example unconvincing.
