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Hopf construction

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Context

Bundles

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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

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

XX*XΣX X \hookrightarrow X\ast X \to \Sigma X

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

Specialized to XX 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[0,1]I \coloneqq [0,1] for the unit interval, regarded as a topological space.

Let X,YX,Y be topological spaces.

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

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

    by the equivalence relation given by

    (x 1,0)(x 2,0),(x 1,1)(x 2,1)x 1,x 2X (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*YX \ast Y is the quotient space

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

    by the equivalence relation

    (x,0,y 1)(x,0,y 2),(x 1,1,y)(x 2,1,y). (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:X×YZ f \colon X \times Y \longrightarrow Z

its Hopf construction is the continuous function

H f:X*YΣZ 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:(x,t,y)(f(x,y),t). H_f \colon (x,t,y) \mapsto (f(x,y), t) \,.

Properties

Homotopy fiber

If XX 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*XΣXX \ast X \to \Sigma X over any point is weakly homotopy equivalent to XX (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 n1X = S^{n-1} a sphere

Proposition

Given a continuous function

f:S n1×S n1S n1 f \colon S^{n-1}\times S^{n-1} \longrightarrow S^{n-1}

the degrees

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

are independent of the choice of xS n1x \in S^{n-1}. The Hopf invariant hh of the Hopf construction H fH_f of ff, def. , is the product of these two:

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

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

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

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

for its dimension as a real vector space. Then the S n1S^{n-1}-sphere may be identified with the subspace of unit norm elements in AA:

S n1{xA;|x| 2=1}. S^{n-1} \simeq \left\{ x \in A \,; {\vert x\vert}^2 = 1 \right\} \,.

Consider then the pairing map

f:S n1×S n1S n1 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 AA:

f:(x,y)xy 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 n1*S n1=(S n1×I×S n1) /{(x,t,y),|x| 2=2t,|y| 2=22t} 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 n1*S n1S 2n1 S^{n-1} \ast S^{n-1} \simeq S^{2n-1}

Similarly, the suspension is parameterized by

ΣS n1=(S n1×I) /{(z,t),|z| 2+(12t) 2=1} \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]I = [0,1] and tIt \in I. This makes manifest that

ΣS n1S n. \Sigma S^{n-1} \simeq S^n \,.

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

(x,y)xy¯ (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 |x| 2=2t{\vert x \vert}^2 = 2t and |y| 2=22t{\vert y\vert}^2 = 2 - 2t then it follows that

|xy¯| 2+(12t) 2 =|x| 2|y| 2+(12t) 2 =2t(22t)+(12t) 2 =1, \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:

S 7 S 4 {(x,t,y),|x| 2=2t,|y| 2=22t} (x,y)zxy¯tt {(z,t),|z| 2+(12t) 2=1}. \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\} } \,.

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

Last revised on February 16, 2019 at 04:18:04. See the history of this page for a list of all contributions to it.