[FOM] Removing Deep Pathology 1

[Colin McLarty]

The circles of the Hopf fibration are just topologically circles, while I
think those intended by the example

> 4)  The partition of 3-space by (infinitely long) non-parallel lines.
> 4a)  "      "     "     "     "  non-parallel circles.

are metric circles.  Anyway the fibers of the Hopf fibration do not lie in
non-parallel planes, since they do not lie in planes at all.

More to Harvey's point, while the Hopf fibration had a surprising
consequence in topology (non-triviality of higher homotopy of spheres), it
is not pathological in any logical sense.  It is explicitly definable by
degree 4 polynomials.

Colin


On Thu, Aug 20, 2015 at 4:36 PM, Hendrik Boom <hendrik at topoi.pooq.com>
wrote:

> On Thu, Aug 20, 2015 at 12:36:45AM +1200, W.Taylor at math.canterbury.ac.nz
> wrote:
> >
> > >just what does nonparallel mean for circles?
> >
> > Lying within non-parallel planes.
>
> My spatial visualisation is challenged here, but with this definition,
> it's possible that the Hopf fibration satisfies the nonparallel
> constraint.  Anyone know for sure?
>
> It does have one straight line -- that's the circle of infinite radius
> I mentioned.  Perhaps that's a deal-breaker.
>
> -- hendrik.
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