Hopf invariant



Homotopy theory

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

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

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see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




For nn \in \mathbb{N} with n>1n \gt 1, consider continuous functions between spheres of the form

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

The homotopy cofiber of ϕ\phi

cofib(ϕ)S nS 2n1D 2n cofib(\phi) \simeq S^n \underset{S^{2n-1}}{\cup} D^{2n}

has ordinary cohomology

H k(cofib(ϕ),){ fork=n,2n; 0 otherwise. H^k(cofib(\phi), \mathbb{Z}) \simeq \left\{ \array{ \mathbb{Z} & for\; k = n, 2n; \\ 0 & otherwise } \right. \,.

Hence for α,β\alpha, \beta generators of the cohomology groups in degree nn and 2n2n (unique up to choice of sign), respectively, there exists an integer h(ϕ)h(\phi) which expresses the cup product square of α\alpha as a multiple of β\beta:

αα=h(ϕ)β. \alpha \cup \alpha = h(\phi) \cdot \beta \,.

This integer h(ϕ)h(\phi) \in \mathbb{Z} is called the Hopf invariant of ϕ\phi (e.g. Mosher-Tangora 86, p. 33).

It depends on the choices made only up to sign. In particular it has a well-defined image [h(ϕ)]𝔽 2=/2[h(\phi)] \in \mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}, and as such it is the Steenrod square

[h(ϕ)]():𝔽 2H n(cofib(ϕ),𝔽 2)Sq nH 2n(cofib(ϕ),𝔽 2)𝔽 2. [h(\phi)] \cdot (-) \;\colon\; \mathbb{F}_2 \simeq H^n(cofib(\phi), \mathbb{F}_2) \stackrel{Sq^n}{\longrightarrow} H^{2n}(cofib(\phi), \mathbb{F}_2) \simeq \mathbb{F}_2 \,.


Generic values

For nn odd, the Hopf invariant necessarily vanishes. For nn even however, then there is a homomorphism

π 2n1(S n) \pi_{2n-1}(S^n) \longrightarrow \mathbb{Z}

whose image contains at least the even integers.

Hopf invariant one

Hence a famous open question in the 1950s was for which maps ϕ\phi one has Hopf invariant one, h(ϕ)=1h(\phi) = 1.

The Hopf invariant one theorem (Adams60) states that the only maps of Hopf invariant one, h(ϕ)=1h(\phi) = 1, are the Hopf constructions on the four real normed division algebras:

Via Sullivan models


By standard results in rational homotopy theory, every continuous function

S 4k1ϕS 2k S^{4k-1} \overset{\phi}{\longrightarrow} S^{2k}

corresponds to a unique dgc-algebra homomorphism

CE(𝔩S 4k1)CE(𝔩ϕ)CE(𝔩S 2k) CE \big( \mathfrak{l}S^{4k-1} \big) \overset{ CE(\mathfrak{l}\phi) }{\longleftarrow} CE \big( \mathfrak{l}S^{2k} \big)

between Sullivan models of n-spheres.

The unique free coefficient of this homomorphism CE(𝔩ϕ)CE(\mathfrak{l}\phi) is the Hopf invariant HI(ϕ)HI(\phi) of ϕ\phi:

Whitehead integral formula

See at Whitehead integral formula and see the references below



See also:

Whitehead’s integral formula

Discussion via differential forms/rational homotopy theory (see also at functional cup product):

Last revised on October 2, 2019 at 14:20:04. See the history of this page for a list of all contributions to it.