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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Observed surface forms (3)

Surface form	Occurrences
Pontryagin–Thom construction	1
Thom isomorphism	1
Thom spaces	1

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 NERFINISHED ⓘ
property	compatible with Whitney sum of bundles ⓘ functorial up to homotopy ⓘ stable under suspension ⓘ
relatedConcept	Chern classes ⓘ Euler class ⓘ Gysin sequence NERFINISHED ⓘ Pontryagin–Thom construction NERFINISHED ⓘ Stiefel–Whitney classes NERFINISHED ⓘ Thom class ⓘ Thom cobordism theory NERFINISHED ⓘ Thom isomorphism NERFINISHED ⓘ 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 ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

René Thom → knownFor → Thom space construction ⓘ

Lev Pontryagin → notableWork → Thom space construction ⓘ

this entity surface form: Pontryagin–Thom construction

Characteristic Classes → hasSubject → Thom space construction ⓘ

this entity surface form: Thom isomorphism

Characteristic Classes → hasSubject → Thom space construction ⓘ

this entity surface form: Thom spaces