Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06
How 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.

Excision for singular homology

Statement

If Z⊆X and Z‾⊆int⁡X(A), then inclusion (X−Z,A−Z)→(X,A) induces isomorphisms Hn(X−Z,A−Z;G)≅Hn(X,A;G) for every n.

Facts & Assumptions

Given: Z‾⊆int⁡X(A).

Proof

technique · direct
1.1

Use the cover {X∖Z‾,int⁡A}: its cover-small quotient complex for (X,A) consists, modulo small chains in A, of precisely the small chains avoiding Z.

givenconstruct
2.1

The two inclusions from their small complexes to the corresponding full relative complexes are chain homotopy equivalences. The quotient identification in step 1.1 therefore induces the stated isomorphism.

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

12 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