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This programme develops the concept of a covering surface.
Module code and title: M335, Studies in pure mathematics M335; 06 29-07-1982 1982 00:25:00 + Show more... Jack Koumi Graham Flegg; Colin Rourke BBC Open University Base surface; Covering surface; Local homeomorphism; Locus The concept of a covering surface has appeared (in disguise) in previous programmes. For example, in TV5, "the locus of the equation w² = z, over an annular region of the z-plane, is itself an annulus. Moreover the two-to-one correspondence between the locus and the z-annulus is a local homeomorphism. The locus is called a "two-sheeted covering surface" of the z-annulus (called the "base surface"). In general, a surface is called a "regular n-sheeted covering surface" if it maps by an n-to-one local homeomorphism onto a base surface. The programme uses several models from previous programmes to point out other familiar examples of covering surfaces. Further examples are developed, using animations, where the covering surface is constructed by appropriate glueing of copies of the base surface. For example, the Mobius band can be thought of as a three-sheeted covering surface of itself (but not a two-sheeted covering surface). The question arises: what surfaces can cover what other surfaces? The possibilities are explored by considering the three topological invariants (Orientability, Euler characteristic, number of boundary components) and asking what can happen to each when we pass from cover to base. The answer, in the form of three theorems, is used to eliminate various possibilities for cover and base. For example, the Euler characteristic of an n-sheeted cover is n times that of the base. Hence, the sphere, characteristic 2, cannot double-cover itself. Finally, The Euler characteristic theorem is generalised to take account of branched covers (those with some branch-points). The three theorems are then used to determine the locus of a cubic equation. The method which students have used in the past to determine loci involves some tricky geometry, algebra and analysis. However, use of the theorems in this case merely requires some simple arithmetic. HOU4206 FOUM152T 978 no