h-cobordism theorem
E265515
The h-cobordism theorem is a fundamental result in differential topology that classifies when two high-dimensional manifolds are diffeomorphic by analyzing the structure of a cobordism between them.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Whitehead torsion | 3 |
| PL h-cobordism theorem | 1 |
| h-cobordism theorem canonical | 1 |
| s-cobordism theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2418313 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: h-cobordism theorem Context triple: [John Milnor, notableWork, h-cobordism theorem]
-
A.
Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
-
B.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
-
C.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
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D.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
-
E.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: h-cobordism theorem Target entity description: The h-cobordism theorem is a fundamental result in differential topology that classifies when two high-dimensional manifolds are diffeomorphic by analyzing the structure of a cobordism between them.
-
A.
Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
-
B.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
-
C.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
D.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
-
E.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in differential topology ⓘ |
| appearsIn | Smale's work on higher-dimensional Poincaré conjecture ⓘ |
| appliesTo | high-dimensional manifolds ⓘ |
| assumption |
compact manifolds
ⓘ
h-cobordism is a cobordism with inclusions homotopy equivalences ⓘ simply connected manifolds ⓘ smooth category ⓘ |
| author | Stephen Smale ⓘ |
| concerns |
cobordism
ⓘ
diffeomorphisms ⓘ h-cobordisms ⓘ smooth manifolds ⓘ |
| conclusion |
cobordism is diffeomorphic to M × [0,1]
ⓘ
h-cobordism is diffeomorphic to a product ⓘ |
| dimensionCondition | dimension at least 5 ⓘ |
| dimensionRestrictionReason | handle-trading arguments require dimension at least 5 ⓘ |
| failsInDimension | dimension 3 ⓘ |
| field |
differential topology
ⓘ
geometric topology ⓘ |
| generalizationOf | Poincaré conjecture in higher dimensions ⓘ |
| hasVariant |
h-cobordism theorem
self-linksurface differs
ⓘ
surface form:
PL h-cobordism theorem
topological h-cobordism theorem ⓘ |
| historicalImportance | central result in the development of high-dimensional manifold theory ⓘ |
| implies | high-dimensional Poincaré conjecture for simply connected manifolds ⓘ |
| influenced |
classification of simply connected manifolds in dimensions ≥ 5
ⓘ
development of s-cobordism theorem ⓘ |
| influencedBy | Morse theory of smooth functions ⓘ |
| involves |
fundamental group
ⓘ
homology groups ⓘ homotopy type ⓘ |
| provedInDecade | 1960s ⓘ |
| provedInYear | 1961 ⓘ |
| relatedConcept |
Morse Theory
ⓘ
surface form:
Morse theory
h-cobordism theorem self-linksurface differs ⓘ
surface form:
Whitehead torsion
handle decomposition ⓘ h-cobordism theorem self-linksurface differs ⓘ
surface form:
s-cobordism theorem
surgery theory ⓘ |
| relates |
diffeomorphism type
ⓘ
homotopy equivalence ⓘ |
| requires |
Morse functions on manifolds
ⓘ
handle cancellation techniques ⓘ |
| standardReference | John Milnor's book "Lectures on the h-Cobordism Theorem" ⓘ |
| subtleInDimension | dimension 4 ⓘ |
| usedFor |
classification of high-dimensional manifolds
ⓘ
proof of the generalized Poincaré conjecture in high dimensions ⓘ surgery theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: h-cobordism theorem Description of subject: The h-cobordism theorem is a fundamental result in differential topology that classifies when two high-dimensional manifolds are diffeomorphic by analyzing the structure of a cobordism between them.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Whitehead torsion
this entity surface form:
Whitehead torsion
this entity surface form:
PL h-cobordism theorem
this entity surface form:
s-cobordism theorem
this entity surface form:
Whitehead torsion