mapping.

Let us note that the cylinder of the $C$-swaddling construction can be ‘cut’ into 2 discs if and only if it is homotopic to a cylinder of which the ‘middle circle’ is constant- in other words as a framing it is homotopic to the constant framing. Then and only then can we ‘pinch closed’ that sphere, turning the cylinder into two discs tangent at a point without loss of information. For elements of the non-trivial homotopy class, this is by definition going to be impossible. Notice that by ‘pinching’ the cylinder into discs, we are separating the backward sweeping tangents and the forward sweeping tangents, which is impossible in the non-trivial homotopy class in which these two families of tangent vectors are one and the same.

But for elements of the trivial homotopy class of $SO(3)$, no such difficulty arises. Stokes’ theorem takes us from $\tau_{\phi}(\gamma)$ to $\tau_{\beta}(\gamma)$. We have proved then the following theorem:

$\tau_{\beta}+4\pi=\tau_{\phi}$

The “$+4\pi$” correction is an idea of Tomotada Ohtsuki’s, to remind us that for ‘minimal’ representatives of $\phi$ and $\beta$ the area $\phi$ covers on $S^{2}$ with the cylinder $C$ (in this case ‘minimal’ would be taken to mean that each “vertical” line 