nLab Hopf construction

Idea

Let $X$ be an H-space. The Hopf construction (Hopf 35) on $X$ is a fibration

$$X \hookrightarrow X\ast X \to \Sigma X$$

whose fiber is $X$, whose base space is the suspension of $X$, and whose total space is the join of $X$ with itself. (Stasheff 70, chapter 1).

Specialized to $X$ the sphere of dimension 0, 1, 3, or 7, the Hopf construction yields the Hopf fibrations. (And by the Hopf invariant one theorem these are the only dimensions for in which spheres are H-spaces.)

Definition

Definition

Write $I \coloneqq [0,1]$ for the unit interval, regarded as a topological space.

Let $X,Y$ be topological spaces.

1. The suspension $\Sigma X$ is the quotient space

   $$\Sigma X \coloneqq (X \times I)_{/\sim}$$

   by the equivalence relation given by

   $$(x_1,0) \sim (x_2,0) \;\,,\;\; (x_1, 1) \sim (x_2, 1) \;\;\; \forall x_1,x_2 \in X$$

2. The join $X \ast Y$ is the quotient space

   $$X \ast Y \coloneqq (X \times I \times Y)_{/\sim}$$

   by the equivalence relation

   $$(x, 0, y_1) \simeq (x,0,y_2) \;\;,\;\; (x_1,1,y) \sim (x_2, 1, y) \,.$$

Definition

Given a continuous function of the form

$$f \colon X \times Y \longrightarrow Z$$

its Hopf construction is the continuous function

$$H_f \colon X \ast Y \longrightarrow \Sigma Z$$

out of the join into the suspension, given in the coordinates of def. by

$$H_f \colon (x,t,y) \mapsto (f(x,y), t) \,.$$

Properties

Homotopy fiber

If $X$ is an “grouplike H-space”, in that it is a topological magma such that left multiplication acts by weak homotopy equivalences, then the homotopy fiber of the Hopf construction $X \ast X \to \Sigma X$ over any point is weakly homotopy equivalent to $X$ (here).

Beware that it may not generally be true that the ordinary fibers of the Hopf construction are weakly homotopy equivalent to the homotopy fibers, see also the discussion of quasifibrations below. But in the classical examples it Happens to be the case, see at Hopf fibration.

Relation to Hopf invariant

Consider $X = S^{n-1}$ a sphere

Proposition

Given a continuous function

$$f \colon S^{n-1}\times S^{n-1} \longrightarrow S^{n-1}$$

the degrees

$$\alpha \coloneqq deg(f(x,-)) \;\;\; \beta \coloneqq deg(f(-,x))$$

are independent of the choice of $x \in S^{n-1}$. The Hopf invariant $h$ of the Hopf construction $H_f$ of $f$, def. , is the product of these two:

$$h(H_f) = \alpha \beta \,.$$

(Mosher-Tangora, exercises to section 4, page 38)

Realization as a quasi-fibration

Beware that Stasheff 70, theorem 1.2 claims that Sugawara claimed that the Hopf construction for any CW H-space is necessarily a quasifibration. But it seems (here) that Sugawara never actually claimed this and also (here) that it is not actually the case.

A different but homotopy-equivalent realization of the Hopf construction, which over grouplike H-spaces is guaranteed to be a quasifibration, is maybe given in Dold-Lashof 59, see also Stasheff 70, theorem 1.4.

Examples

Hopf fibrations

When $X$ is a sphere that is an $H$-space, namely, one of the groups $S^0 = \mathbb{Z}/2$ the group of order 2, $S^1 = U(1)$ the circle group, the 3-sphere special unitary group $S^3 = SU(2)$; or the 7-sphere $S^7$ with its Moufang loop structure, then the Hopf construction produces the four Hopf fibrations:

1. $S^0 \hookrightarrow S^1 \to S^1$ – real Hopf fibration
2. $S^1 \hookrightarrow S^3 \to S^2$ – complex Hopf fibration
3. $S^3 \hookrightarrow S^7 \to S^4$ – quaternionic Hopf fibration
4. $S^7 \hookrightarrow S^{15} \to S^8$ – octonionic Hopf fibration

In detail, let $A \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$ be one of the real normed division algebras and write

$$n \coloneqq dim_{\mathbb{R}}(A) \in \{1,2,4,8\}$$

for its dimension as a real vector space. Then the $S^{n-1}$-sphere may be identified with the subspace of unit norm elements in $A$:

$$S^{n-1} \simeq \left\{ x \in A \,; {\vert x\vert}^2 = 1 \right\} \,.$$

Consider then the pairing map

$$f \colon S^{n-1}\times S^{n-1} \longrightarrow S^{n-1}$$

which is the restriction to these unit norm elements of the product in $A$:

$$f \colon (x,y) \mapsto x \cdot y$$

This is well defined by the very property that for normed division algebras the norm is multiplicative.

Accordingly, the join of two such spheres is naturally parameterized as follows

$$S^{n-1}\ast S^{n-1} = (S^{n-1}\times I \times S^{n-1})_{/\sim} \simeq \left\{ (x,t,y) \,, {\vert x \vert}^2 = 2t \,,\; {\vert y\vert}^2 = 2 - 2t \right\}$$

which makes manifest that

$$S^{n-1} \ast S^{n-1} \simeq S^{2n-1}$$

Similarly, the suspension is parameterized by

$$\Sigma S^{n-1} = (S^{n-1}\times I)_{/\sim} \simeq \left\{ (z,t) \,,\; {\vert z \vert}^2 + (1 - 2t)^2 = 1 \right\}$$

where we take $I = [0,1]$ and $t \in I$. This makes manifest that

$$\Sigma S^{n-1} \simeq S^n \,.$$

Moreover, in this parameterization the Hopf construction, def. , which is given by

$$(x,y) \mapsto x \cdot \overline{y}$$

manifestly gives the Hopf fibration map.

Notice that it is again the multiplicativity of the norm in division algebras which makes this work: if ${\vert x \vert}^2 = 2t$ and ${\vert y\vert}^2 = 2 - 2t$ then it follows that

$$\begin{aligned} {\vert x \cdot \overline{y}\vert}^2 + (1- 2t)^2 & = {\vert x \vert}^2 {\vert y \vert}^2 + (1-2t)^2 \\ & = 2t (2-2t) + (1 - 2t)^2 \\ & = 1 \end{aligned} \,,$$

hence that indeed we have a well-defined map like so:

$$\array{ S^7 & \longrightarrow & S^4 \\ \left\{ (x,t,y) \,, {\vert x \vert}^2 = 2t \,,\; {\vert y\vert}^2 = 2 - 2t \right\} &\stackrel{{(x,y) \mapsto z \coloneqq x \cdot \overline{y}}\atop{t \mapsto t}}{\longrightarrow}& \left\{ (z,t) \,,\; {\vert z \vert}^2 + (1 - 2t)^2 = 1 \right\} } \,.$$

See also

• Hopf construction in homotopy type theory

References

The original sources are

• Heinz Hopf, Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension, Fund. Math. 25: 427–440 (1935) (Euclid)

• George Whitehead, On the homotopy groups of spheres and rotation groups, Annals of Mathematics. Second Series 43 (4): 634–640, (1942) (JSTOR)

• Albrecht Dold, Richard Lashof, Principal quasifibrations and fibre homotopy equivalence of bundles, Illinois J. Math. Volume 3, Issue 2 (1959), 285-305 (euclid:1255455128)

Review inclides

• Jim Stasheff, chapter 1 in H-Spaces from a Homotopy point of view, Lecture Notes in Mathematics Volume 161 1970

Textbook accounts include

• Robert Mosher, Martin Tangora, p. 38 of Cohomology Operations and Application in Homotopy Theory, Harper and Row (1968) (pdf)

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 10.6 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

See also

• Guillermo Moreno, Hopf construction map in higher dimensions (arXiv:math/0404172)

• Feza Guersey, Chia-Hsiung Tze, (4b.2) in On the role of Division, Jordan and Related algebras in Particle Physics

Discussion of the situation in parameterized homotopy theory includes

• A. L. Cook, M.C. Crabb, Fiberwise Hopf structures on sphere bundles, J. London Math. Soc. (2) 48 (1993) 365-384 (pdf)

• Kouyemon Iriye, Equivariant Hopf structures on a sphere, J. Math. Kyoto Univ. Volume 35, Number 3 (1995), 403-412 (Euclid)

See also

• Wikipedia, Hopf construction