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.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The monodromy right action on a covering fibre and its equivalent left-action convention
Definition
Fix a covering , a basepoint , and . For , define as the endpoint of the unique lift of beginning at (Existence and uniqueness of path lifts through a covering map). Endpoint homotopy invariance makes this well defined (The endpoint of a lifted path depends only on its endpoint-fixed homotopy class). With the library's traversal-order product this is a right action; the corresponding left action is (Left group actions, transitive actions, and faithful actions).
Depends on
Used by
- Endpoint monodromy of an unordered configuration loop as a permutation of the labels Definition
- The two-point unordered cover of the plane and the monodromy of a half turn Example
- Deck transformations of a connected covering correspond to cosets in the subgroup normalizer Lemma
- Every subgroup acts on the universal cover with a connected quotient covering that realizes it Lemma
- Monodromy acts by fibre bijections, and its orbits are the intersections of path components with the fibre Proposition
- For a nonempty path-connected total space, a covering fibre is in bijection with the right cosets of the induced fundamental-group subgroup Theorem
- 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
- The Riemann surface of the logarithm is the complex plane over the punctured plane via exp Theorem
Dependency tree · two levels
19 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)