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.
A CW complex with no cells in adjacent dimensions has zero cellular boundary
Statement
If has no cells in adjacent dimensions, every cellular differential is zero; consequently .
Facts & Assumptions
Given: A CW complex with no two nonzero cellular chain groups in consecutive degrees.
Proof
For every , either or is zero, so .
Thus cellular homology equals the chain group in each degree; Cellular homology computes singular homology transfers this calculation to singular homology.
Depends on
Used by
- A point and projective-line check of source-first convolution and opposite composition Example
- Cellular homology and ring-independent groups of complex projective space Example
- Unequal-dimensional proper pushforwards carry the summed-dimension Ext/Borel–Moore degree Example
- Integral cohomology ring of complex projective space Lemma
- Mod-two cohomology rings of complex projective spaces Lemma
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, Theorem 2.35 (standard reference, not scraped)