proof of Scott-Wiegold conjecture

Suppose the conjecture were false. Then we have some wCp*Cq*Cr with N(w)=Cp*Cq*Cr. Let a, b, c denote the of w onto Cp, Cq, Cr 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<θ<360 we say that a spin through θ consists of a unit vector, u3 together with the rotation of 3 through the angle θ anticlockwise about u. In we have a single spin through the angle 0 and a single spin through 360. Thus the set of spins (usually denoted Spin(3)) naturally has the topologyMathworldPlanetmath of a 3-sphere.

The spin through θ about a unit vector u has the same underlying rotation as the spin through 360-θ about -u. Hence there are precisely two spins corresponding to each rotation of 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 spin about u composed with a 20 spin about u is a 350 spin about -u (not a 10 spin about u which would be at the other end of the 3-sphere).

Let n denote the unit vector (0,0,1). Fix an arc, I, on the unit sphere connecting n and -n. Let t be a vector on this arc. Let u be an arbitrary unit vector. We may define a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ϕt,u:F{a,b,c}Spin(3) by:

ϕt,u:a the spin through (p-12)360p (or 180 if p=2) about n

ϕt,u:b the spin through (q-12)360q (or 180 if q=2) about t

ϕt,u:c the spin through (r-12)360r (or 180 if r=2) about u

(Here F{a,b,c} denotes the free groupMathworldPlanetmath on a,b,c).

So ϕt,u(a), ϕt,u(b) and ϕt,u(c) are spins of between 120 and 180, all having non-trivial underlying rotations.

Let 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 ϕ:Cp*Cq*CrSO(3) induced by ϕ. If ϕt,u(w~) has a trivial underlying rotation for some t and u, then N(w) will only contain elements in the kernel of ϕ. In particular, we would have a,b,cN(w). So we may assume we have a map:


which maps (t,u) to the unit vector corresponding to ϕt,u(w~).

By we have h(n,Ru)=Rh(n,u) for any rotation R about n. Thus h(n,_):S2S2 maps latitudes to latitudes (possibly rotating them and / or moving them up or down).

Also h(n,n)=-n, as ϕn,n(a), ϕn,n(b) and ϕn,n(c) are spins of between 120 and 180 anticlockwise about n, so the sum of the angles will be greater than 360. Similarly one may that h(n,-n)=n. Thus, as h(n,_) maps latitudes to latitudes, it must be homotopicMathworldPlanetmathPlanetmath to a reflection of S2.

Again by we have h(-n,Ru)=Rh(-n,u) for all rotations R about n. Hence h(-n,_):S2S2 also maps latitudes to latitudes.

Further, h(-n,n)=n and h(-n,-n)=-n. Thus h(-n,_) is homotopic to the .

But h gives a homotopyMathworldPlanetmath from h(n,_) to h(-n,_), yielding the desired contradictionMathworldPlanetmathPlanetmath.

Title proof of Scott-Wiegold conjecture
Canonical name ProofOfScottWiegoldConjecture
Date of creation 2013-03-22 18:30:31
Last modified on 2013-03-22 18:30:31
Owner whm22 (2009)
Last modified by whm22 (2009)
Numerical id 5
Author whm22 (2009)
Entry type Proof
Classification msc 20E06