Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Deck transformations and the deck-transformation group of a covering
Definition
For a covering , a deck transformation is an isomorphism over , so (Maps and isomorphisms of covering spaces over a fixed base). Deck transformations form the deck group under composition, and this group acts on by evaluation (Group and abelian group, Left group actions, transitive actions, and faithful actions).
Depends on
Used by
- Regular coverings Definition
- The quotient ℝ→ℝ/ℤ is a covering with integer translations as deck transformations Example
- Rational transfer identifies a finite regular cover with deck invariants Lemma
- On a connected covering space, a deck transformation is determined by one point and the deck action is free Proposition
- For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group Theorem
- Ordered configuration spaces cover the unordered ones regularly with deck group Sₙ 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
Dependency tree · two levels
10 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)