nLab Hopf invariant



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



Paths and cylinders

Homotopy groups

Basic facts




In ordinary cohomology

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

(1)f:S 2n1S n. f \;\colon\; S^{2n-1} \longrightarrow S^n \,.

The homotopy cofiber of ff (the attaching space induced by ff)

C fS nS 2n1D 2n C_f \;\coloneqq\; S^n \underset{S^{2n-1}}{\cup} D^{2n}

has ordinary cohomology

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

Hence for α n,β 2n\alpha_n, \beta_{2n} generators of the cohomology groups in degree nn and 2n2n (unique up to choice of sign), respectively, there exists an integer HI(f)HI(f) which expresses the cup product square of α n\alpha_n as a multiple of β 2n\beta_{2n}:

(2)α nα n=HI(f)β 2n. \alpha_n \cup \alpha_n \;=\; HI(f) \cdot \beta_{2n} \,.

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

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

[HI(f)]():𝔽 2H n(C f;𝔽 2)Sq nH 2n(C f;𝔽 2)𝔽 2. [HI(f)] \cdot (-) \;\colon\; \mathbb{F}_2 \;\simeq\; H^n \big( C_f; \, \mathbb{F}_2 \big) \stackrel{Sq^n}{\longrightarrow} H^{2n} \big( C_f; \, \mathbb{F}_2 \big) \;\simeq\; \mathbb{F}_2 \,.

In generalized cohomology

Here is a more abstract picture of the Hopf invariant in abstract homotopy theory (following SS21):

Let EE be a multiplicative cohomology theory, assumed to vanish in degree 2n12n - 1

(3)E˜(S 2n1)π 2n1(E)0. \widetilde E \big( S^{2n-1} \big) \,\simeq\, \pi_{2n-1}(E) \,\simeq\, 0 \,.

We write E nΩ Σ nE E_n \;\coloneqq\; \Omega^\infty \Sigma^n E for its classifying spaces, hence for the component spaces of its representing spectrum.

(For the case of ordinary cohomology E=HE = H \mathbb{Z} we have E nK(,n)E_n \,\simeq\, K(\mathbb{Z},n) an Eilenberg-MacLane space.)

Now let

S 2n1fS n S^{2n-1} \overset{f}{\longrightarrow} S^n

be a map (1). Then its EE-Hopf invariant “is” the following homotopy pasting diagram of pointed homotopy types (see also at e-invariant is Todd class of cobounding (U,fr)-manifold):

homotopy pasting diagram exhibiting the Hopf invariant
from SS21
  • This yields the homotopy filling the full bottom part of the diagram above; and by the universal property of the bottom homotopy-pushout this corresponds equivalently to a dashed morphism S 2nE 2nS^{2n} \to E_{2n}, labeled by some class

    κπ 0(E)E˜(S 0)Σ 2nE˜(S 2n)=[S 2nE 2n]; \kappa \,\in\, \pi_0(E) \,\simeq\, \widetilde E(S^0) \overset{\Sigma^{2n}}{\longrightarrow} \widetilde E(S^{2n}) \, = \, \big[ S^{2n} \longrightarrow E_{2n} \big] \,;
  • finally, by the homotopy-pushout property of the total rectangle, this class κ\kappa also labels the total homotopy filling the full diagram.

We see that:

In the case that the map ff is one the classical Hopf fibrations, the attaching space above is a projective space (by the discussion at cell structure of projective spaces) and the choice of homotopy cc is the choice of an orientation in EE-cohomology theory to second stage. Specifically:

homotopy pasting diagram exhibiting orientation and the Hopf invariant
from SS21

Moreover, the de-composition of this pasting diagram exhibits the homotopy Whitehead integral/functional cup product-formula for the Hopf invariant:

homotopy pasting diagram exhibiting the homotopy Whitehead integral
from SS21


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

A famous open question in the 1950s was which maps ff (1) have Hopf invariant one, namely [HI(f)]=1[HI(f)] = 1 (2).

The Hopf invariant one theorem (Adams60) states that the only maps of Hopf invariant one, [HI(f)]=1[HI(f)] = 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 4k1fS 2k S^{4k-1} \overset{f}{\longrightarrow} S^{2k}

corresponds to a unique dgc-algebra homomorphism

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

between Sullivan models of n-spheres.

The unique free coefficient of this homomorphism CE(𝔩f)CE(\mathfrak{l}f) is the Hopf invariant HI(f)HI(f) of ff:

Whitehead integral formula

See at Whitehead integral formula and see the references below



See also:

And see the references at Hopf invariant one for more, in particular for the formulation via topological K-theory.

Whitehead’s integral formula

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

Last revised on April 3, 2023 at 11:11:29. See the history of this page for a list of all contributions to it.