Post by Gary on Mar 6, 2017 1:22:21 GMT
Vasco,
In the more elaborate example beginning in paragraph 4, we appear to be asked to bake a cake with a scattering of preimages. As one layer is rolled onto another, preimages are added (projected onto the plane) and counted as the number of new layers lying above a point in the image set. Perhaps we can think of the image set as embedded in a layer of waxed paper (the plane) below the rolled layers (including the original disc layer). That would explain the numbers of layers lying over p: one over inner disc, two over outer ring, three over black ring.
I am unable to follow the final paragraph of the section. Suppose we pick a p from the outer ring, as in the example. Then there must be a preimage in the layer from the top of the hat and another preimage in the layer derived from the side of the extrusion. If we flatten the layers, I would expect them to have the same orientation under projection of the little loops down onto the plane, leading to $\nu[\Gamma, p] = 2$, not 0. How do the two preimages acquire reversed orientation?
Gary
In the more elaborate example beginning in paragraph 4, we appear to be asked to bake a cake with a scattering of preimages. As one layer is rolled onto another, preimages are added (projected onto the plane) and counted as the number of new layers lying above a point in the image set. Perhaps we can think of the image set as embedded in a layer of waxed paper (the plane) below the rolled layers (including the original disc layer). That would explain the numbers of layers lying over p: one over inner disc, two over outer ring, three over black ring.
I am unable to follow the final paragraph of the section. Suppose we pick a p from the outer ring, as in the example. Then there must be a preimage in the layer from the top of the hat and another preimage in the layer derived from the side of the extrusion. If we flatten the layers, I would expect them to have the same orientation under projection of the little loops down onto the plane, leading to $\nu[\Gamma, p] = 2$, not 0. How do the two preimages acquire reversed orientation?
Gary
