Hirzebruch signature theorem

E587785

mathematical 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.

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Observed surface forms (1)

Surface form	Occurrences
signature theorem	1

Statements (45)

Predicate	Object
instanceOf	mathematical theorem ⓘ
appliesTo	4k-dimensional manifolds ⓘ compact manifolds ⓘ oriented manifolds ⓘ smooth manifolds ⓘ
assumes	compactness of the manifold ⓘ orientation on the manifold ⓘ smooth structure on the manifold ⓘ
codomain	integers ⓘ
consequence	constraints on intersection forms of 4k-manifolds ⓘ topological invariance of certain Pontryagin numbers ⓘ
defines	L-genus as characteristic number ⓘ
dimensionCondition	dimension is a multiple of 4 ⓘ
domain	oriented cobordism ring ⓘ
expresses	L-polynomial in terms of Pontryagin classes ⓘ signature of a manifold as a polynomial in Pontryagin classes ⓘ
field	algebraic topology ⓘ differential topology ⓘ global analysis ⓘ
formulaType	characteristic number formula ⓘ
generalizationOf	Gauss–Bonnet theorem for 2-dimensional case (in spirit) ⓘ
gives	ring homomorphism from oriented cobordism ring to integers via signature ⓘ
hasInvariant	Hirzebruch L-polynomial NERFINISHED ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	signature is a cobordism invariant for oriented manifolds ⓘ
inspired	development of index theory ⓘ
involves	intersection form on middle-dimensional cohomology ⓘ rational cohomology ⓘ
mathematicsSubjectClassification	57R20 ⓘ 57R75 ⓘ
namedAfter	Friedrich Hirzebruch NERFINISHED ⓘ
relatedTo	Atiyah–Singer index theorem NERFINISHED ⓘ Hirzebruch–Riemann–Roch theorem NERFINISHED ⓘ Riemann–Roch theorem NERFINISHED ⓘ
relatesConcept	L-genus ⓘ Pontryagin classes NERFINISHED ⓘ characteristic classes ⓘ signature of a manifold ⓘ
states	signature of a smooth compact oriented 4k-manifold equals its L-genus ⓘ
usedIn	classification of 4-manifolds ⓘ cobordism theory ⓘ study of smooth structures on manifolds ⓘ
uses	Pontryagin classes of the tangent bundle ⓘ rational Pontryagin classes ⓘ
yearProved	1950s ⓘ

Referenced by (2)

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

Friedrich Hirzebruch → knownFor → Hirzebruch signature theorem ⓘ

Hirzebruch–Riemann–Roch theorem → relatedTo → Hirzebruch signature theorem ⓘ

this entity surface form: signature theorem