Geometric 3-manifolds
From Manifold Atlas
Contents |
1 Introduction
Let a group act on a manifold by homeomorphisms.
A -manifold is a manifold with a -atlas, that is, a collection of homeomorphisms onto open subsets of such that all coordinate changes
are restrictions of elements of .
Fix a basepoint and a chart with . Let be the universal covering. These data determine the developing map
that agrees with the analytic continuation of along each path, in a neighborhood of the path's endpoint.
If we change the initial data and , the developing map changes by composition with an element of .
If , analytic continuation along a loop representing gives a chart that is comparable to x_0g_\sigmaG\phi_0^\sigma=g_\sigma\phi_0$. The mapis a group homomorphism and is called the holonomy of .
If we change the initial data and , the holonomy homomorphisms changes by conjugation with an element of .
A -manifold is complete if the developing map is surjective.
2 Construction and examples
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3 Invariants
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4 Classification/Characterization
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5 Further discussion
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6 References
This page has not been refereed. The information given here might be incomplete or provisional. |