I was once criticized by students for amending an example I had given them when I came to work it myself. Riemann did the same thing in a big way when he changed the domain and range of implicit functions from the complex plane to Riemann surfaces. Over time a great edifice was built over Riemann surfaces while the original topic stagnated. This may be because to treat it, to understand branches, one had to use cuts which were regarded as arbitrary and inelegant. And if by some means branches could be got without using cuts the arbitrariness would still be there because cuts and boundaries of branches are the same thing.
   To use or not use cuts is not just a matter of taste. Previous attempts to understand branches may have failed because of a mistaken assumption that a given branch-point has a unique permutation on branches associated with it. You have to introduce a system of cuts and for each cut determine an associated permutation. While the need for cuts cannot be got round there is a way to make their arbitrariness acceptable and to complete the theory. One can quantify over it. This is shown in detail in the other files.

   Copyright July2004 conesetter. Quotation with acknowledgement is permitted.