Hopf invariant

E679320

homotopy invariant integer-valued invariant topological invariant

The Hopf invariant is a topological integer-valued invariant that classifies certain continuous maps between spheres, playing a central role in homotopy theory and the study of higher-dimensional linking.

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Observed surface forms (1)

Surface form	Occurrences
Hopf degree theorem	1

Statements (48)

Predicate	Object
instanceOf	homotopy invariant ⓘ integer-valued invariant ⓘ topological invariant ⓘ
appearsIn	Adams’s solution of the Hopf invariant one problem ⓘ obstruction theory ⓘ rational homotopy theory ⓘ study of H-spaces ⓘ
appliesTo	continuous maps between spheres ⓘ maps S^{2n-1} → S^n ⓘ
centralIn	Hopf invariant one problem NERFINISHED ⓘ
constraint	Hopf invariant one maps exist only in dimensions 1, 2, 4, and 8 ⓘ
context	CW-complex mapping cones ⓘ maps between spheres of odd dimension domain ⓘ
definedOn	homotopy classes of maps ⓘ
definedUsing	cellular decomposition of mapping cone ⓘ cohomology operations ⓘ cup product in cohomology ⓘ
field	algebraic topology ⓘ homotopy theory ⓘ
generalization	Massey products NERFINISHED ⓘ secondary cohomology operations ⓘ
hasSpecialCase	Hopf invariant of the Hopf fibration S^7 → S^4 equals 1 ⓘ Hopf invariant of the Hopf fibration S^{15} → S^8 equals 1 ⓘ Hopf invariant of the Hopf map S^3 → S^2 equals 1 ⓘ
hasVariant	mod p Hopf invariant NERFINISHED ⓘ reduced Hopf invariant ⓘ stable Hopf invariant ⓘ
implies	existence of higher-dimensional linking phenomena ⓘ
namedAfter	Heinz Hopf NERFINISHED ⓘ
property	additive under composition in certain contexts ⓘ homotopy invariant of maps ⓘ
relatedTo	Adams spectral sequence NERFINISHED ⓘ Hopf fibration NERFINISHED ⓘ J-homomorphism NERFINISHED ⓘ Whitehead product NERFINISHED ⓘ cohomology cup product ⓘ complex numbers ⓘ linking number ⓘ normed division algebras ⓘ octonions ⓘ quaternions ⓘ real numbers ⓘ stable homotopy groups of spheres ⓘ
usedFor	classification of certain homotopy classes of maps between spheres ⓘ distinguishing non-homotopic maps with same degree ⓘ study of higher-dimensional linking ⓘ
usedToProve	nontriviality of certain homotopy groups of spheres ⓘ
valueType	integer ⓘ

Referenced by (2)

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

Heinz Hopf → notableWork → Hopf invariant ⓘ

this entity surface form: Hopf degree theorem