Universal covering spaces

Depends on

• Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
• Simply connected topological spaces

Used by

• Bonnet-Myers fundamental group is finite Corollary
• Every Riemann surface is a quotient of a simply connected model Corollary
• ℝ→ℝ/ℤ is a universal covering Corollary
• Based cellular chains of a universal cover as finite free right group-ring modules Definition
• Right action on universal-cover chains Definition
• Spherical, parabolic and hyperbolic universal-covering types Definition
• The based handle chain complex over the fundamental group ring Definition
• The suspension foliation of a representation of the fundamental group Definition
• A complex torus has a lattice of parabolic deck translations Example
• A genus-two compact surface gives a cocompact Fuchsian group Example
• Annulus and punctured disc have hyperbolic universal covers Example
• A contractible relative group-ring complex with a pi-one isomorphism detects a homotopy equivalence Lemma
• A universal covering of a Riemann surface inherits a unique complex structure Lemma
• Closed hyperbolic three-manifold groups contain no Z-squared subgroup Lemma
• Compatible arbitrary pi fence reduction Lemma
• Finite surface normal forms, Jordan disks, and torsion control Lemma
• Simple lifted caps avoid the original essential loop and a fixed intrinsic neighbourhood Lemma
• The covering of a leaf associated with the holonomy kernel exists Lemma
• The handle complex of an h-cobordism is contractible over the group ring, with an explicit contraction Lemma
• The Poincaré sphere is a nontrivial integral homology sphere Lemma
• The space of local geodesics, its length metric, and the covering criterion for local isometries Lemma
• Suspension holonomy is the germ of the represented monodromy action Proposition
• A space admitting a universal covering is semilocally simply connected Theorem
• Cartan hadamard Theorem
• Complete, simply connected, locally CAT(0) length spaces are CAT(0) Theorem
• Every nonempty path-connected locally path-connected semilocally simply connected space has a universal cover Theorem
• For a path-connected locally path-connected base, a universal cover maps uniquely over the base to every connected covering, and any two universal covers are uniquely isomorphic Theorem

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Allen Hatcher, Algebraic Topology, §1.3 (standard reference, not scraped)
• J. Peter May, A Concise Course in Algebraic Topology, Ch. 3 (standard reference, not scraped)
• Marco Gualtieri, MAT1300 Week 4 Term 2, §1.6 (standard reference, not scraped)