Abstract
Covering spaces are introduced and studied, in particular for topological groups.
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Notes
- 1.
The material in this chapter will not really be needed before Chap. 6. See MA X, no 3.
- 2.
This assumption is not really necessary since every “sufficiently small” open subset is contained in a connected open subset over which (X, p) is trivial.
- 3.
An equivalence relation R on a space X is said to be open if the R-saturation of every open subset is open. If, moreover, the graph of R is closed and if X is separable, the same holds for X / R.
- 4.
The similarity with equality \( d(g'g''z)/dz = d(g'g'' z)/d(g''z) \times d(g''z)/dz\) is not entirely a coincidence.
- 5.
They do not appear to be in the published literature. The construction of \(\widetilde{G}\) as the set of pairs \((g,\alpha ) \) with composition law (2.7.23) has been known to experts for a long time, but for experts, of which I am supposed to be one, this construction obviously leads to a genuine covering of the group G. It assumes that a topology is established on the set of these pairs \((g,\alpha )\), that its compatibility with the composition law (2.7.23) holds, and finally that so does the “local triviality” condition of the coverings for the map p. Hence, these “trivial verifications” are usually omitted as it is reckoned that, if a central extension of G by \(\mathbb Z\) is (abstractly) constructed, Providence will provide the topology that will turn it into a universal covering of G. This expectation is of course fully justified in hindsight, which explains why the detailed construction is little more than a tedious exercise for “beginners”.
- 6.
This result could have been proved beforehand since \(B_+\) is simply connected and it would seem that it could have then easily provided a direct construction of \(\widetilde{G}\). Unfortunately, we would have needed to express the composition law of \(\widetilde{G}\) in terms of the decomposition considered...
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Godement, R. (2017). Simply Connected Spaces and Groups. In: Introduction to the Theory of Lie Groups. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-54375-8_2
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DOI: https://doi.org/10.1007/978-3-319-54375-8_2
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