Origin of the idea of covering maps

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# Origin of the idea of covering maps

Submitted by halb on Tue, 09/20/2005 - 11:55

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Hi,

I was wondering what the origin of the idea of covering spaces and covering maps are. Would it be correct to say that the idea is some kind of generalization of (or that its origin lies in the idea of) Riemann surfaces and many valued functions?

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## Re: Origin of the idea of covering maps

I would agree with you. Historically, covering maps and many other topics in topology have their origins in complex analysis. I can think of two early occurrences of the idea of covering spaces: 1. Maxwell's study of multiple-valued electric and magnetic potentials in multiply-connected regions. 2. Wirtinger's study of branched covers over curves which was arose from the study of functions of two complex variables such as the solution of the cubic equation. In both of these cases, it is pretty clear that the inspiration for introducing covering spaces comes from trying to generalize well-known constructions of complex analysis to higher-dimensional spaces --- Maxwell generalized the notion of branch cut and Wirtinger that of branch point. Maxwell and his contemporaries were aware of the fact that harmonic functions can be seen as a generalization of complex analytic functions (or, conversely, complex analysis can be seen as the study of harmonic functions in two dimensions). Wirtinger was motivated by trying to apply the techniques of Riemann surfaces to problems involving more than one complex varianle.

So I would say that the idea of the covering map definitely has its origin as a generalization of the idea of a Rieman surface. To be sure, it would be an number of years after the pioneering work of Maxwell, Wirtinger and others before a formal definition of "covering map" could be formulated; to formulate such a definition, one needs first to be able to frame rigorous definitions of such terms as "topological space", "open set", "simplicial complex" and "manifold". However, when this happened, the process of generalization and abstraction which led to the concept of a covering space from that of a Riemann surface did not stop. The concepts of bundle and of sheaf were successively distilled from the concept of covering space by these processes.