proof of Scott-Wiegold conjecture
Suppose the conjecture were false. Then we have some with . Let , , denotethe of onto , , respectively. Then, , are all non-trivial as otherwise would becontained in the kernel of one of the .
For we say that a spin through consists of a unit vector, together with therotation of through the angle anticlockwise about .In we have a single spin through the angle anda single spin through . Thus the set of spins(usually denoted Spin(3)) naturally has the topology of a3-sphere.
The spin through about a unit vector has thesame underlying rotation as the spin through about . Hence there are precisely two spinscorresponding to each rotation of 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 spin about composed with a spin about is a spin about (not a spin about which would be at the other end of the 3-sphere).
Let denote the unit vector . Fix an arc, ,on the unit sphere connecting and . Let be a vector on this arc. Let be an arbitraryunit vector. We may define a homomorphismby:
the spin through (or if )about
the spinthrough (or if) about
the spinthrough (or if) about
(Here denotes the free group on ).
So , and are spins of between and, all having non-trivial underlying rotations.
Let be a word in representing , such that occur in it Mod times, Mod times and Mod times, respectively.
We have a homomorphism induced by . If has a trivial underlying rotation forsome and , then will only containelements in the kernel of . In particular, we would have . Sowe may assume we have a map:
which maps to the unit vector corresponding to.
By we have for any rotation about . Thus maps latitudes to latitudes (possibly rotating themand / or moving them up or down).
Also , as, and are spins of between and anticlockwise about , so the sum of theangles will be greater than . Similarly one may that . Thus, as maps latitudes to latitudes, it must be homotopic to areflection of .
Again by we have for all rotations about .Hence also maps latitudes tolatitudes.
Further, and . Thus is homotopic to the .
But gives a homotopy from to , yielding the desired contradiction
.