nLab Hopf invariant one



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

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models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




In algebraic topology, Hopf invariant one refers to homotopy classes of continuous functions between spheres of the form

ϕ:S 2n1S n, \phi \;\colon\; S^{2n-1} \longrightarrow S^n \,,

hence to elements in the homotopy groups of spheres

[ϕ]π 2n1(S n), [\phi] \in \pi_{2n-1}(S^n) \,,

whose Hopf invariant is equal to one:

h(ϕ)=1. h(\phi) = 1 \,.

Often this is regarded not in the integers but in its quotient to the cyclic group of order 2 /2\mathbb{Z}/2, where it hence says that the actual integer value is an odd number.

The Hopf invariant one problem, once a famous open problem, solved in Adams 60, is the classification of maps for which Hopf invariant is indeed equal to one.


Relation to H-space structure and normed division algebras

The existence of an element of Hopf invariant one in π 2n1(S n)\pi_{2n-1}(S^n) is equivalent to the existence of an H-space structure on S n1S^{n-1}.

A celebrated theorem due to (Adams 60, introducing and using the Adams spectral sequence) states that maps of Hopf invariant one correspond precisely to the the Hopf constructions on the four normed division algebras (see also at Hurwitz theorem): the real Hopf fibration, the complex Hopf fibration, the quaternionic Hopf fibration and the octonionic Hopf fibration.

(Adams 60)


Stable homotopy theory emerged as a distinct branch of algebraic topology with Adams‘ introduction of his eponymous spectral sequence and his spectacular conceptual use of the notion of stable phenomena in his solution to the Hopf invariant one problem.

(Elmendorf-Kriz-May 95)

exceptional spinors and real normed division algebras

AA\phantom{AA}spin groupnormed division algebra\,\, brane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})A\phantom{A} \mathbb{R} the real numberssuper 1-brane in 3d
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})A\phantom{A} \mathbb{C} the complex numberssuper 2-brane in 4d
6=5+16 = 5+1Spin(5,1)Spin(5,1) \simeq SL(2,H)A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) {\simeq}SL(2,O)A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/type II string


The original proof that the only maps of Hopf invariant one are the Hopf constructions on the four normed division algebras is due to

and made use of the classical Adams spectral sequence.


Comments on the impact of this proof on the development of stable homotopy theory:

Another proof that instead uses topological K-theory, Adams operations and the Atiyah-Hirzebruch spectral sequence was given in

see also Adams 66, Sections 7, 8.

Review of the Hopf invariant one problem via Adams operations in topological K-theory:

Review using the BP-Adams-Novikov spectral sequence includes

Last revised on November 16, 2023 at 02:49:45. See the history of this page for a list of all contributions to it.