DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

K handle core cocore attaching region and belt sphere

Definition

For integers $0 \leq k \leq n$ , the standard $n$ -dimensional $k$ -handle is $D^{k} \times D^{n - k}$ . Its core is $D^{k} \times {0}$ , its cocore is ${0} \times D^{n - k}$ , its attaching region is $S^{k - 1} \times D^{n - k}$ , and its attaching sphere is $S^{k - 1} \times {0}$ . The outgoing region is $D^{k} \times S^{n - k - 1}$ and the belt sphere is ${0} \times S^{n - k - 1}$ . Here $D^{j}$ is the closed unit disk, $D^{0}$ is a point, and $S^{- 1} = \emptyset$ . For $n = 0$ both boundary regions are empty.

Used by

Attaching a smooth handle with corner rounding Definition
Local morse sublevel pair is a handle pair Lemma
Relative homology of the standard handle pair Lemma

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Nicolaescu, An Invitation to Morse Theory (standard reference, not scraped)
Benedetti, Lectures on Differential Topology (standard reference, not scraped)