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It is well-known that, on a
finitely connected, noncompact, Riemann surface, the complex-dimension of
the space of all analytic differentials modulo the space of exact analytic
differentials is the first Betti number of the underlying surface, and hence its
real-dimension twice the first Betti number. Further, it is well-known that
the group of units of the algebra of analytic functions on such a surface
modulo the subgroup of exponential functions is a free Abelian group whose
rank is again the first Betti number of the underlying surface. Thus, in each
case, the analytic obstruction on the surface fully dualizes the continuous
obstruction.
Interestingly, this is not the case on a finitely connected, noncompact,
nonorientable Klein surface; for example, in the case of the first problem, the
real-dimension is twice the first Betti number minus one. In the second problem, the
group in question is isomorphic to the direct sum of the two element group and the
free Abelian group whose rank is the first Betti number, of the underlying
space, minus one. These calculations are first made using sheaf theory, in
§2. Integration theory is then applied, §3, to elucidate the reason that this
curious defect occurs. Application is then made, using integration, to a mixed
harmonic—analytic obstruction problem in §4. Finally, the Dirichlet deficiency
of the analogue of the standard algebra on compact, nonorientable, Klein
surfaces— with boundary—is computed. Again the defect of minus one occurs.
Throughout, the reason why this defect occurs in the nonorientable case is of prime
concern.
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