# proof of Scott-Wiegold conjecture

Suppose the conjecture were false. Then we have some $w\in C_{p}*C_{q}*C_{r}$ with $N(w)=C_{p}*C_{q}*C_{r}$. Let $a$, $b$, $c$ denote the of $w$ onto $C_{p}$, $C_{q}$, $C_{r}$ respectively. Then $a$, $b$, $c$ are all non-trivial as otherwise $N(w)$ would be contained in the kernel of one of the .

For $0^{\circ}<\theta<360^{\circ}$ we say that a spin through $\theta$ consists of a unit vector, $\vec{u}\in\mathbb{R}^{3}$ together with the rotation of $\mathbb{R}^{3}$ through the angle $\theta$ anticlockwise about $\vec{u}$. In we have a single spin through the angle $0^{\circ}$ and a single spin through $360^{\circ}$. Thus the set of spins (usually denoted Spin(3)) naturally has the topology of a 3-sphere.

The spin through $\theta$ about a unit vector $\vec{u}$ has the same underlying rotation as the spin through $360^{\circ}-\theta$ about $-\vec{u}$. Hence there are precisely two spins corresponding to each rotation of $\mathbb{R}^{3}$ about the origin.

is well defined on spins as you can compose the underlying rotations and continuity determines which of the two spins is the correct result. For example a $350^{\circ}$ spin about $\vec{u}$ composed with a $20^{\circ}$ spin about $\vec{u}$ is a $350^{\circ}$ spin about $-\vec{u}$ (not a $10^{\circ}$ spin about $\vec{u}$ which would be at the other end of the 3-sphere).

Let $\vec{n}$ denote the unit vector $(0,0,1)$. Fix an arc, $I$, on the unit sphere connecting $\vec{n}$ and $-\vec{n}$. Let $\vec{t}$ be a vector on this arc. Let $\vec{u}$ be an arbitrary unit vector. We may define a homomorphism $\phi_{\vec{t},\vec{u}}\colon F_{\{a,b,c\}}\to{\rm Spin}(3)$ by:

$\phi_{\vec{t},\vec{u}}\colon a\mapsto$ the spin through $(\frac{p-1}{2})\frac{360^{\circ}}{p}$ (or $180^{\circ}$ if $p=2$) about $\vec{n}$

$\phi_{\vec{t},\vec{u}}\colon b\mapsto$ the spin through $(\frac{q-1}{2})\frac{360^{\circ}}{q}$ (or $180^{\circ}$ if $q=2$) about $\vec{t}$

$\phi_{\vec{t},\vec{u}}\colon c\mapsto$ the spin through $(\frac{r-1}{2})\frac{360^{\circ}}{r}$ (or $180^{\circ}$ if $r=2$) about $\vec{u}$

(Here $F_{\{a,b,c\}}$ denotes the free group on $a,b,c$).

So $\phi_{\vec{t},\vec{u}}(a)$, $\phi_{\vec{t},\vec{u}}(b)$ and $\phi_{\vec{t},\vec{u}}(c)$ are spins of between $120^{\circ}$ and $180^{\circ}$, all having non-trivial underlying rotations.

Let $\tilde{w}$ be a word in $F_{\{a,b,c\}}$ representing $w$, such that $a,b,c$ occur in it $1$ Mod $(2p)$ times, $1$ Mod $2q$ times and $1$ Mod $(2r)$ times, respectively.

We have a homomorphism $\phi^{\prime}:C_{p}*C_{q}*C_{r}\to SO(3)$ induced by $\phi$. If $\phi_{\vec{t},\vec{u}}(\tilde{w})$ has a trivial underlying rotation for some $\vec{t}$ and $\vec{u}$, then $N(w)$ will only contain elements in the kernel of $\phi^{\prime}$. In particular, we would have $a,b,c\notin N(w)$. So we may assume we have a map:

 $h\colon I\times S^{2}\to S^{2}$

which maps $(\vec{t},\vec{u})$ to the unit vector corresponding to $\phi_{\vec{t},\vec{u}}(\tilde{w})$.

By we have $h(\vec{n},R\vec{u})=Rh(\vec{n},\vec{u})$ for any rotation $R$ about $\vec{n}$. Thus $h(\vec{n},\_)\colon S^{2}\to S^{2}$ maps latitudes to latitudes (possibly rotating them and / or moving them up or down).

Also $h(\vec{n},\vec{n})=-\vec{n}$, as $\phi_{\vec{n},\vec{n}}(a)$, $\phi_{\vec{n},\vec{n}}(b)$ and $\phi_{\vec{n},\vec{n}}(c)$ are spins of between $120^{\circ}$ and $180^{\circ}$ anticlockwise about $\vec{n}$, so the sum of the angles will be greater than $360^{\circ}$. Similarly one may that $h(\vec{n},-\vec{n})=\vec{n}$. Thus, as $h(n,\_)$ maps latitudes to latitudes, it must be homotopic to a reflection of $S^{2}$.

Again by we have $h(-\vec{n},R\vec{u})=Rh(-\vec{n},\vec{u})$ for all rotations $R$ about $\vec{n}$. Hence $h(-\vec{n},\_)\colon S^{2}\to S^{2}$ also maps latitudes to latitudes.

Further, $h(-\vec{n},\vec{n})=\vec{n}$ and $h(-\vec{n},-\vec{n})=-\vec{n}$. Thus $h(-\vec{n},\_)$ is homotopic to the .

But $h$ gives a homotopy from $h(\vec{n},\_)$ to $h(-\vec{n},\_)$, yielding the desired contradiction.

Title proof of Scott-Wiegold conjecture ProofOfScottWiegoldConjecture 2013-03-22 18:30:31 2013-03-22 18:30:31 whm22 (2009) whm22 (2009) 5 whm22 (2009) Proof msc 20E06