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单词 ProofOfScottWiegoldConjecture
释义

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 denotethe of w onto Cp, Cq, Cr respectively. Thena, b, c are all non-trivial as otherwise N(w) would becontained in the kernel of one of the .

For 0<θ<360 we say that a spin throughθ consists of a unit vector, u3 together with therotation of 3 through the angle θ anticlockwise about u.In we have a single spin through the angle 0 anda single spin through 360. Thus the set of spins(usually denoted Spin(3)) naturally has the topologyMathworldPlanetmath of a3-sphere.

The spin through θ about a unit vector u has thesame underlying rotation as the spin through 360-θabout -u. Hence there are precisely two spinscorresponding to each rotation of 3 about the origin.

is well defined on spins as you can compose theunderlying rotations and continuity determines which of the twospins is the correct result. For example a 350 spin aboutu composed with a 20 spin about u is a350 spin about -u (not a 10 spin aboutu 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. Lett be a vector on this arc. Let u be an arbitraryunit vector. We may define a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathϕt,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 spinthrough (q-12)360q (or 180 ifq=2) about t

ϕt,u:c the spinthrough (r-12)360r (or 180 ifr=2) about u

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

So ϕt,u(a), ϕt,u(b) andϕt,u(c) are spins of between 120 and180, 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. Ifϕt,u(w~) has a trivial underlying rotation forsome t and u, then N(w) will only containelements in the kernel of ϕ. In particular, we would have a,b,cN(w). Sowe may assume we have a map:

h:I×S2S2

which maps (t,u) to the unit vector corresponding toϕt,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 themand / or moving them up or down).

Also h(n,n)=-n, asϕn,n(a), ϕn,n(b) andϕn,n(c) are spins of between 120 and180 anticlockwise about n, so the sum of theangles 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 areflection 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 tolatitudes.

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.

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更新时间:2025/5/4 16:42:18