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.
Relative connecting homomorphism on cycles
Definition
For a relative cycle represented by with , define the connector in the pair sequence by This fixes the convention in which the quotient map is the third arrow of the short exact sequence of chains.
Depends on
Used by
- Relative homology of a disk and its boundary Example
- Relative homology of an interval and its endpoints Example
- Coordinate-ball classes identify local homology stalks Lemma
- Long exact sequence of a triple in singular homology Lemma
- One handle changes relative homology in one degree only Lemma
- The cyclic cover retracts onto the lifted flower and has a deck-equivariant spine model Lemma
- Universal-cover boundaries, maps and homotopies respect the right group-ring action Lemma
- Well-definedness of the relative connector Lemma
- Serre transgression agrees with the relative connecting construction Proposition
- The unreduced module fits an exact sequence with the reduced module Proposition
Dependency tree · two levels
3 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, §2.1 (standard reference, not scraped)