Some Non-Abelian Problems on Compact Riemann Surfaces
Robert C. Gunning · Rice Research Repository (Rice University) · 1968
A good deal of tlte classical function theory of compact Riemalili surfaces is of an essentially abelian nature, liaviiig to do with complex analytic properties of flat coinplex line bundles.Problems do arise, tho~~gli, which are reaIly non-abelian in character, having to do with cornplex analytic properties of flat complex vector bundles.These problems lead in quite interesting but relatively little explored directions.The aim of this lecture is merely to show how the beginnings of the non-abelian theory can be developed in a manner paralleling a familiar development of the abelian theory.First, to establish a background for the discussioll, a few definitions should be recalled.Consider a cotnplex analytic manifold M of complex dimension n, and let U = {U,) be a coverilig of M by open coordinate neigl~borltoods U,.A one-cocycle of the covering U with coefficients in an arbitrary abstract group G is a collection of elements ( , / , E G, indexed by ordered pairs (U,, Ufl) of sets of the covering LT for which U, f? Up # @, such that ( , , = I and that = 1 whenever U, f? U,{ n U, # @.The set of all such one-cocycIes will be denoted by Z1(U, G).Two one-cocycles (5,/,) and (tio) of Zi(U,G) are called equivalent if there is a collection of elements TI, E G, indexed by the sets U, of the covering U, such that ti,, = r ~, ~, ~i l , ~~ whenever U,nUp#@; it is easy to see that this is an eqtlivalence relation in the usual sense of the term.The set of equivalence classcs is called tlte first cohomology set of the covering LI witli coefficients in the group G, and will be denoted by H1(U,G).The coltomology as thus defined depends on the choice of the coveri~ig LI, and not just or1 the space M ; but it can be shown that for well-behaved coveriiigs the coho~nologies are in natural one-to-one correspondences.(See for instance the discussioli of Leray's theorem in [2].)Hereafter it will be assumed that only such well-behaved coverings are considered, and the common coltomology set will be denoted by H1(M, G) atid called the first coliomology of the space M witli coefficients in the group G.If the group G is abelian, it is clear that the set of cocycles form an abelian group; the group operations are compatible with the equivalence relation, so that the coltomology set H1(M, G) is also an abelian group.39