Thom cobordism theory

E627198

branch of algebraic topology cobordism theory homotopy-theoretic classification theory

Thom cobordism theory is a foundational branch of algebraic topology developed by René Thom that classifies manifolds up to cobordism using homotopy-theoretic and characteristic class methods.

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Statements (48)

Predicate	Object
instanceOf	branch of algebraic topology ⓘ cobordism theory ⓘ homotopy-theoretic classification theory ⓘ
appliesTo	smooth manifolds ⓘ topological manifolds ⓘ
associatedWith	complex Thom spectrum MU ⓘ oriented Thom spectrum MSO ⓘ symplectic Thom spectrum MSp NERFINISHED ⓘ unoriented Thom spectrum MO NERFINISHED ⓘ
centralConcept	bordism group ⓘ cobordism ring ⓘ normal bundle ⓘ stable normal bundle ⓘ
characterizedBy	Pontryagin–Thom construction NERFINISHED ⓘ
classifies	oriented manifolds up to cobordism ⓘ smooth manifolds up to cobordism ⓘ unoriented manifolds up to cobordism ⓘ
defines	complex cobordism ring ⓘ oriented cobordism ring ⓘ unoriented cobordism ring ⓘ
developedBy	René Thom NERFINISHED ⓘ
field	algebraic topology ⓘ
goal	classification of manifolds up to cobordism equivalence ⓘ
historicalResult	computation of unoriented cobordism ring ⓘ description of oriented cobordism in terms of characteristic numbers ⓘ
implies	structure of graded ring on cobordism groups ⓘ
influenced	generalized cohomology theories ⓘ stable homotopy theory ⓘ
introduces	Thom space NERFINISHED ⓘ Thom spectrum NERFINISHED ⓘ
provides	homotopy-theoretic description of cobordism groups ⓘ
relatedTo	bordism ⓘ complex cobordism theory ⓘ generalized homology theories ⓘ surgery theory ⓘ
relates	bordism to characteristic classes ⓘ bordism to stable homotopy theory ⓘ cobordism groups to homotopy groups of Thom spectra ⓘ
studies	cobordism classes of manifolds ⓘ
uses	Chern classes NERFINISHED ⓘ Euler class NERFINISHED ⓘ Pontryagin classes NERFINISHED ⓘ Pontryagin–Thom collapse map NERFINISHED ⓘ Stiefel–Whitney classes NERFINISHED ⓘ Thom spaces NERFINISHED ⓘ Thom spectra NERFINISHED ⓘ characteristic classes ⓘ homotopy theory ⓘ

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René Thom → knownFor → Thom cobordism theory ⓘ