Alphabeta MathDefinitionDefinition: 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 , the standard -dimensional -handle is . Its core is , its cocore is , its attaching region is , and its attaching sphere is . The outgoing region is and the belt sphere is . Here is the closed unit disk, is a point, and . For both boundary regions are empty.
Used by
- Adjacent-index handles with zero intersection do not cancel Counterexample
- Algebraic intersection one with three geometric points Counterexample
- Attaching a smooth handle with corner rounding Definition
- Attaching-belt intersection matrix of adjacent-index handles Definition
- Dual handle decomposition Definition
- Dual surgery sphere Definition
- Framed embedded surgery sphere Definition
- Geometrically cancelling adjacent handle pair Definition
- Handle decomposition relative to the incoming boundary Definition
- Handle slide of one k handle over another Definition
- p-surgery on a smooth m-manifold Definition
- Surgery trace cobordism Definition
- The based handle chain complex over the fundamental group ring Definition
- The middle-handle intersection matrix of an h-cobordism Definition
- A cancelling one-two handle pair on a surface Example
- A cancelling zero-one handle pair Example
- A handle slide realizes an elementary row operation Example
- A one-handle between distinct manifold components is a boundary connected sum Lemma
- Algebraic cancellation does not yet give geometric cancellation Lemma
- Belt-sphere complements in low handle levels preserve the fundamental group Lemma
- Elimination lemma: trading a handle for a handle two indices higher Lemma
- Formal-immersion homotopies extend over a subcritical handle Lemma
- Group-labelled Whitney tricks realize the diagonalized handle complex Lemma
- h-cobordisms admit two-index normal form presentations Lemma
- Handle attachments are relative cell attachments up to homotopy Lemma
- Handle boundary coefficients are attaching-belt intersection numbers Lemma
- Homology lemma: a handle-basis class is realized by a sphere meeting the belt sphere once Lemma
- Isotopic attaching embeddings give diffeomorphic handle attachments Lemma
- Local morse sublevel pair is a handle pair Lemma
- One handle changes relative homology in one degree only Lemma
- One transverse intersection gives the standard local cancelling model Lemma
- Relative homology of the standard handle pair Lemma
- Standard handle admits an adapted Morse function Lemma
- The dual handle retraction onto the cocore, with the outgoing region carried onto the belt sphere Lemma
- The group-labelled homology lemma realizes group-ring handle bases by isotopy Lemma
- The outgoing boundary of a handle attachment trades the disk factors Lemma
- The standard complementary pair fills a ball Lemma
- Zero- and one-handles are eliminated in a simply connected h-cobordism Lemma
- Realization of prescribed Whitehead torsion by h-cobordisms Proposition
- The handle chain complex computes singular homology Proposition
…and 2 more results.
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)