Lifting curves
Here is a picture of a helix H and an annulus A. You should think
of the helix as being infinite in both directions, although we only
have space to show a few loops. Vertical projection gives a map
p : H -> A. We use the points marked in red as basepoints for H
and A.
Let S be a sector in A as shown at the bottom. Then the preimage
p-1(S) is a disjoint union of connected pieces, which we
call Ti. Let pi be p considered as a map from
Ti to S. It should be clear from the picture that
pi : Ti -> S is a homeomorphism.
Suppose we have a closed curve u : S1 -> A as shown
below, and we want to lift it to a loop in H. We start by lifting
a short section of the curve, which we have marked in red. There
are infinitely many wys to do this, of which three are shown. Only
one of the lifts (shown in red) starts at the basepoint; we choose
this one.
The section of the curve that we have already lifted is shown in
blue; we now try to extend this lift to the next section of the
curve, which is shown in red at the bottom. There are again
infinitely many ways to lift the red section, of which we have shown
three. Only one of these (shown in red) joins up with our lift of
the first section of the curve; we choose this one.
The section of the curve that we have already lifted is shown in
blue; we now try to extend this lift to the next section of the
curve, which is shown in red at the bottom. There are again
infinitely many ways to lift the red section, of which we have shown
three. Only one of these (shown in red) joins up with our lift of
the first section of the curve; we choose this one.
The section of the curve that we have already lifted is shown in
blue; we now try to extend this lift to the next section of the
curve, which is shown in red at the bottom. There are again
infinitely many ways to lift the red section, of which we have shown
three. Only one of these (shown in red) joins up with our lift of
the previous sections of the curve; we choose this one.
The section of the curve that we have already lifted is shown in
blue; we now try to extend this lift to the last section of the
curve, which is shown in red at the bottom. There are again
infinitely many ways to lift the red section, of which we have shown
three. Only one of these (shown in red) joins up with our lift of
the previous sections of the curve; we choose this one.
We now have a lift of the whole curve to H, shown in blue. However,
the lifted curve does not close up; it ends in a different place to
where it started. The gap between them is shown in red. The end is
one sheet above the beginning, which mean that the winding number of
the original curve is 1.
