nLab
Hopf invariant one

Contents

Context

Homotopy theory

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

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

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 \,.

Properties

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 nS^n.

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)

History

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

Lorentzian
spacetime
dimension
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)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1)Spin(9,1) {\simeq}SL(2,O)A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/type II string

References

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

  • Frank Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. (The Annals of Mathematics, Vol. 72, No. 1) 72 (1): 20–104, (1960) (JSTOR)

and made use of the classical Adams spectral sequence.

Review includes

Further comments on the impact of this proof on the development of stable homotopy theory includes

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

This is reproduced for instance in

Quick review includes

Review using the BP-Adams-Novikov spectral sequence includes

Last revised on September 17, 2018 at 06:54:22. See the history of this page for a list of all contributions to it.