Thom space construction

E627199

concept in algebraic topology topological construction

The Thom space construction is a fundamental operation in algebraic topology that associates a topological space to a vector bundle, playing a central role in cobordism theory and characteristic classes.

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All labels observed (5)

Label	Occurrences
Pontryagin–Thom construction	2
Thom isomorphism	2
Thom spaces	2
Thom space	1
Thom space construction canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T6901171 — 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: Thom space construction
Context triple: [René Thom, knownFor, Thom space construction]

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. h-cobordism theorem
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.
C. Thom–Mather stratification
Thom–Mather stratification is a refined notion of stratification in differential topology that imposes strong regularity and control conditions on how smooth strata fit together, generalizing and strengthening Whitney stratifications.
D. Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
E. Hirzebruch signature theorem
The Hirzebruch signature theorem is a fundamental result in differential topology that expresses the signature of a smooth, compact, oriented 4k-dimensional manifold as a polynomial in its Pontryagin classes.
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: Thom space construction
Target entity description: The Thom space construction is a fundamental operation in algebraic topology that associates a topological space to a vector bundle, playing a central role in cobordism theory and characteristic classes.

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. h-cobordism theorem
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.
C. Thom–Mather stratification
Thom–Mather stratification is a refined notion of stratification in differential topology that imposes strong regularity and control conditions on how smooth strata fit together, generalizing and strengthening Whitney stratifications.
D. Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
E. Hirzebruch signature theorem
The Hirzebruch signature theorem is a fundamental result in differential topology that expresses the signature of a smooth, compact, oriented 4k-dimensional manifold as a polynomial in its Pontryagin classes.
F. None of above. chosen

Statements (50)

Predicate	Object
instanceOf	concept in algebraic topology ⓘ topological construction ⓘ
appliesTo	complex vector bundles ⓘ oriented vector bundles ⓘ real vector bundles ⓘ stable vector bundles ⓘ topological vector bundles ⓘ
centralRoleIn	cobordism theory ⓘ construction of complex cobordism MU ⓘ construction of oriented cobordism MSO ⓘ construction of spin cobordism MSpin ⓘ construction of unoriented cobordism MO ⓘ
definitionUses	collapse map ⓘ disk bundle of a vector bundle ⓘ one-point compactification ⓘ quotient space construction ⓘ sphere bundle of a vector bundle ⓘ
field	algebraic topology ⓘ cobordism theory ⓘ differential topology ⓘ homotopy theory ⓘ stable homotopy theory ⓘ
hasInput	base space of a vector bundle ⓘ total space of a vector bundle ⓘ vector bundle ⓘ
hasOutput	Thom space ⓘ pointed topological space ⓘ
namedAfter	René Thom ⓘ
property	compatible with Whitney sum of bundles ⓘ functorial up to homotopy ⓘ stable under suspension ⓘ
relatedConcept	Chern classes ⓘ Euler class ⓘ Gysin sequence ⓘ Pontryagin–Thom construction ⓘ linked to: Pontryagin–Thom collapse map Stiefel–Whitney classes ⓘ Thom class ⓘ Thom cobordism theory ⓘ Thom isomorphism ⓘ linked to: Thom space construction Thom spectrum ⓘ orientation of a vector bundle ⓘ
usedFor	associating a topological space to a vector bundle ⓘ constructing orientation classes ⓘ defining Gysin maps ⓘ defining Thom isomorphism in cohomology ⓘ defining Thom spectra ⓘ defining cobordism theories ⓘ defining generalized cohomology theories ⓘ studying characteristic classes ⓘ studying embeddings of manifolds ⓘ

How these facts were elicited