One approach is to understand this as part of the geometry of universal covering spaces. Such spaces are contractiblesimply connected (even sometimes contractible), but may have finite groups acting on them sufficiently "nicely" (and there's the catch) so that the quotient inherits the finite group as its fundamental group. In fact if the long exact sequence of a fibration (of homotopy groups) applies, you can force a fundamental group to coincide with a discrete group as fibre.
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One approach is to understand this as part of the geometry of universal covering spaces. Such spaces are contractible, but may have finite groups acting on them sufficiently "nicely" (and there's the catch) so that the quotient inherits the finite group as its fundamental group. In fact if the long exact sequence of a fibration (of homotopy groups) applies, you can force a fundamental group to coincide with a discrete group as fibre. |
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