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.
Oriented cellular chain group
Definition
For a CW complex and abelian group , set and for . By Relative homology of consecutive CW skeleta, an orientation of each characteristic -disk selects the corresponding direct-summand copy of .
Depends on
Used by
- Based cellular chains of a universal cover as finite free right group-ring modules Definition
- Cellular boundary from three consecutive skeleta Definition
- Cellular homology Definition
- Incidence number of two CW cells Definition
- Integral cohomology ring of a closed orientable surface Example
- Integral cohomology ring of complex projective space Example
- Cellular boundary coefficients are the signed trajectory counts Lemma
- Cellular homology of the one-polygon surface model Lemma
- Homological AHSS exact couple from the skeletal filtration Lemma
- Integral surface cup pairing from the oriented polygon Lemma
- Local coordinate cup products generate top relative cohomology Lemma
- Mod-two cohomology rings of complex projective spaces Lemma
- Rational cohomology scalar comparison on compact smooth manifolds Lemma
- The AHSS E-one page is cellular cochains with theory coefficients Lemma
- The AHSS first differential is the cellular coboundary Lemma
- The cyclic cover retracts onto the lifted flower and has a deck-equivariant spine model Lemma
- Cellular boundary is the incidence degree matrix Theorem
- The Morse complex is chain isomorphic to the handle cellular complex Theorem
Dependency tree · two levels
9 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, Section 2.2 (standard reference, not scraped)