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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 ?

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Any two rotations of $S^2$ with different axes will compose to another rotation (with a... different different axis). So there'll be two antipodal points $x_1, x_2$ with $f(x_i)=g(x_i)$. –  Micah Nov 9 '12 at 23:55
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Indeed -- there's a nice formula for the axis of rotation of the product of two rotations of $S^2$. It's given by a formula that uses quaternionic multiplication. –  Ryan Budney Nov 10 '12 at 0:01

2 Answers 2

up vote 1 down vote accepted

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").

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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.

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