LemmaStatement: AI-generatedProof: AI-generated

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.

Exact range of a cofiber connectivity bound

PDF p. 4: equivalence in degrees at most three

Statement

If the cofiber of $f : A \to B$ in $D (Z)$ is supported in degrees $> 3$ , meaning its cohomology vanishes in every degree at most three, then $H^{i} (f)$ is an isomorphism for every $i \leq 3$ .

Facts & Assumptions

Given: $f$ and the stated cofiber bound.

Goal: Prove the isomorphism range.

[F1]

The mapping-cone long exact sequence used in Cofiber bounds imply low-degree cohomology isomorphisms contains $H^{i - 1} (C) \to H^{i} (A) \to H^{i} (B) \to H^{i} (C)$ .

Proof

1.1F1given

For each $i \leq 3$ , both $i - 1$ and $i$ are at most three, so both adjacent groups in [F1] vanish.

2.1F1givenstep 1.1∎

Exactness makes $H^{i} (A) \to H^{i} (B)$ injective and surjective. Thus the stated range follows.

Depends on

Cofiber bounds imply low-degree cohomology isomorphisms

Used by · 1

Low-degree consequences of a weight-one comparison

Dependency tree · 3 levels

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