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quaternionic Hopf fibration

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Topology

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

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

Introduction

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Extra stuff, structure, properties

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Analysis Theorems

topological homotopy theory

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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

The quaternionic Hopf fibration is the Hopf fibration induced by the quaternions, hence it is the fibration

S 3 S 7 p S 4 \array{ S^3 &\hookrightarrow& S^7 \\ && \downarrow^{\mathrlap{p_{\mathbb{H}}}} \\ && S^4 }

of the 7-sphere over the 4-sphere with fiber the 3-sphere, which is induced via the Hopf construction from the product operation

×()() \mathbb{H} \times \mathbb{H} \stackrel{(-)\cdot (-)}{\longrightarrow} \mathbb{H}

on the quaternions, or else from

× ×()() 1 \mathbb{H} \times \mathbb{H}^{\times} \stackrel{(-)\cdot (-)^{-1}}{\longrightarrow} \mathbb{H}

to match standard conventions.

This means that if S 7S^7 is regarded as the unit sphere {(x,y)||x| 2+|y| 2=1}\{(x,y) | {\vert x\vert}^2 + {\vert y\vert}^2 = 1\} in ×\mathbb{H}\times \mathbb{H} and S 4S^4 is regarded as the quaternionic projective space, then pp is given (on points (x,y)(x,y) with y0y \neq 0) simply by

p :(x,y)[x;y]=[x/y;1], p_{\mathbb{H}} \colon (x,y) \mapsto [x;y] = [x/y; 1] \,,

Properties

SO(3)SO(3)-Equivariant structure

Since the automorphism group of the quaternions, as an \mathbb{R}-algebra, is the special orthogonal group SO(3)SO(3)

Aut ()SO(3) \mathrm{Aut}_{\mathbb{R}}(\mathbb{H}) \simeq SO(3)

acting by rotation of the imaginary quaternions, via the Hopf construction it follows that the 7-sphere and 4-sphere inherit SO(3)SO(3)-actions under which the quaternionic Hopf map is equivariant.

Notice that this means that SO(3)SO(3) acts on S 7S^7 here diagonally on the two copies of the imaginary octonions in S 7S^7 \hookrightarrow \mathbb{H} \oplus \mathbb{H} (as opposed to, say, via any one of the embeddings SO(3)SO(8)SO(3) \hookrightarrow SO(8) and the following canonical action of SO(8)SO(8) on S 7 8S^7 \hookrightarrow \mathbb{R}^8).

(see also Cook-Crabb 93)

Class in the homotopy groups of spheres

The quaternionic Hopf fibration gives an element in the 7th homotopy group of the 4-sphere

[p ]π 7(S 4)×(/12) [p_{\mathbb{H}}] \in \pi_7(S^4) \simeq \mathbb{Z} \times (\mathbb{Z}/12)

and in fact it is a generator of the non-torsion factor in this group.

Stably, i.e. as a generator for the stable homotopy groups of spheres in degree 74=37-4 = 3, the quaternionic Hopf map becomes a torsion generator

[p ]π 3 S/24,. [p_{\mathbb{H}}] \in \pi_3^S \simeq \mathbb{Z}/24 \,,.

Class in equivariant stable homotopy theory

Fix a finite subgroup GSO(3)G \hookrightarrow SO(3) which does not come from SO(2)SO(3)SO(2) \hookrightarrow SO(3) – i.e. not a cyclic group, but one of the dihedral groups or else the tetrahedral group or octahedral group or icosahedral group (by the ADE classification).

Regard both S 7S^7 and S 4S^4 as pointed topological G-spaces via the SO(3)SO(3)-action induced via automorphisms of the quaternions, as above. Write

Σ G S 7,Σ G S 4GSpectra \Sigma^\infty_G S^7, \Sigma^\infty_G S^4 \in G Spectra

for the corresponding equivariant suspension spectra.

Notice that if we took trivial GG, then in the stable homotopy category

[Σ S 7,Σ S 4]/24 [\Sigma^\infty S^7, \Sigma^\infty S^4] \simeq \mathbb{Z}/24

by the above. In contrast:1

Proposition

In GG-equivariant homotopy theory this becomes a non-torsion group, i.e.

[Σ G S 7,Σ G S 4] G [\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G \simeq \mathbb{Z} \oplus \cdots

with the quaternionic Hopf fibration, regarded as a GG-equivariant map, representing a non-torsion element.

Proof

First use the Greenlees-May decomposition which says that for any two GG-equivariant spectra X,YX,Y and writing π (X),π (Y)\pi_\bullet(X), \pi_\bullet(Y) for their equivariant homotopy groups, organized as Mackey functors Hπ n H(X)H \mapsto \pi_n^H(X) for all subgroups HGH \subset G, then the canonical map

[X,Y] GnHom [G](π n(X),π n(Y)) [X,Y]_G \longrightarrow \underset{n}{\oplus} Hom_{\mathcal{M}[G]}(\pi_n(X), \pi_n(Y))

is rationally an isomorphism.

With this we are reduced to showing that there exists nn \in \mathbb{Z} and a morphism of Mackey functors of equivariant homotopy groups π n(Σ G S 7)π n(Σ G S 4)\pi_n(\Sigma^\infty_G S^7) \to \pi_n(\Sigma^\infty_G S^4) which is not a torsion element in the abelian hom-group of Mackey functors.

To analyse this, we use the tom Dieck splitting which says that the equivariant homotopy groups of equivariant suspension spectra Σ G X\Sigma^\infty_G X contain a direct summand which is simply the ordinary stable homotopy groups of the naive fixed point space X HX^H:

π n H(Σ G X)π n(Σ (X H)). \pi_n^H(\Sigma^\infty_G X) \simeq \pi_n(\Sigma^\infty (X^H)) \oplus \cdots \,.

Now observe that the fixed points of the SO(3)SO(3)-action on the quaternionic Hopf fibration that we are considering is just the real Hopf fibration:

(p ) SO(3)=p :S 1S 1 (p_{\mathbb{H}})^{SO(3)} = p_{\mathbb{R}} \;\colon\; S^1 \longrightarrow S^1

since SO(3)SO(3) acts transitively on the quaternionic quaternions and fixes the real quaternions. By our assumption that GSO(3)G \subset SO(3) does not come through SO(2)SO(3)SO(2) \hookrightarrow SO(3) it follows that this statment is still true for GG:

(p ) G=p :S 1S 1. (p_{\mathbb{H}})^{G} = p_{\mathbb{R}} \;\colon\; S^1 \longrightarrow S^1 \,.

But the real Hopf fibration defines a non-torsion element in π 0 S\pi_0^S \simeq \mathbb{Z}.

In conclusion then, at n=1n = 1 and H=GH = G we find that the GG-equivariant quaternionic Hopf fibration contributes a non-torsion element in

Hom Ab(π 1 G(Σ G S 7),π 1 G(Σ G S 4)) Hom_{Ab}(\pi_1^G(\Sigma^\infty_G S^7), \pi_1^G(\Sigma^\infty_G S^4))

which appears as a non-torsion element in

Hom [G](π 1(Σ G S 7),π 1(Σ G S 4)) Hom_{\mathcal{M}[G]}( \pi_1(\Sigma^\infty_G S^7), \pi_1(\Sigma^\infty_G S^4) )

and hence in [Σ G S 7,Σ G S 4] G[\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G.

See also at equivariant stable cohomotopy

References

  • Reiko Miyaoka, The linear isotropy group of G 2/SO(4)G_2/SO(4), the Hopf fibering and isoparametric hypersurfaces, Osaka J. Math. Volume 30, Number 2 (1993), 179-202. (Euclid)

Discussion 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)

Discussion in homotopy type theory is in


  1. The proof of prop. profited from Charles Rezk, who suggested here that the reduction to fixed points will make the real Hopf fibration give a non-torsion contribution, and from David Barnes who amplified the use of the Greenless-May splitting theorem.

Last revised on August 14, 2018 at 10:07:43. See the history of this page for a list of all contributions to it.