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.