Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings

Depends on

• Continuity of a map of topological spaces at a point and globally
• Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
• Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
• The disjoint union (coproduct) $\bigsqcup_i X_i$ with the final topology of the canonical injections: a set is open exactly when each of its traces is

Used by

• A connected covering of a locally path-connected simply connected space is one-sheeted and trivial Corollary
• Covering homomorphisms of Lie groups Definition
• Lifts of maps, paths, and homotopies through a covering map Definition
• Maps and isomorphisms of covering spaces over a fixed base Definition
• Regular coverings Definition
• The pullback of a covering space along a continuous map Definition
• Universal covering spaces Definition
• Cut locus of a point on a flat circle Example
• Cut locus on a flat rectangular torus from the Dirichlet cell Example
• Pulling a covering back to an evenly covered open set gives a trivial covering Example
• The maps [x]↦[mx] on ℝ/ℤ are m-sheeted coverings for m≥1 Example
• The two-point unordered cover of the plane and the monodromy of a half turn Example
• Trivial coverings are products with a discrete fibre Example
• The distance from p is smooth on m minus p False statement
• Connected covers of smooth manifolds have a canonical smooth structure Lemma
• Covering homotopies lift by finite local strips Lemma
• Disjoint coordinate neighbourhoods evenly cover the unordered configuration space Lemma
• Every subgroup acts on the universal cover with a connected quotient covering that realizes it Lemma
• For a nonempty path-connected locally path-connected semilocally simply connected space, the path-class projection is a covering map Lemma
• Rational transfer identifies a finite regular cover with deck invariants Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• Two high relative cell layers have free homotopy bases and their cellular boundary matrix Lemma
• A based morphism between connected coverings exists exactly when the induced subgroups are included Proposition
• A composite of covering maps is a covering when the outer covering is finite-sheeted Proposition
• A local homeomorphism from a nonempty compact Hausdorff space to a connected Hausdorff space is a finite-sheeted covering Proposition
• Covering maps are surjective local homeomorphisms with discrete fibres Proposition
• Covering spaces are stable under restriction, finite products, and pullback Proposition
• Local path-connectedness lifts and descends along covering maps Proposition
• The cardinality of a covering fibre is locally constant and is constant on a connected base Proposition
• The germ projection of a complete analytic function is a covering map Remark
• Degree of a proper holomorphic map of Riemann surfaces Theorem
• Existence and uniqueness of path lifts through a covering map Theorem
• For a finite-sheeted covering, the total space is compact exactly when the base is compact Theorem
• Ordered configuration spaces cover the unordered ones regularly with deck group Sₙ Theorem
• p:ℝ→ℝ/ℤ is a covering map with translated interval sheets Theorem
• The orbit map of a covering-space action is a covering, with the acting group equal to the deck group when the total space is path-connected Theorem
• Two lifts from a connected space that agree at one point agree everywhere Theorem

Dependency tree · two levels

11 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)