nLab
7-sphere

Contents

Context

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

Manifolds and cobordisms

Contents

Idea

The sphere of dimension 7.

This is one of the parallelizable spheres, as such corresponds to the octonions among the division algebras, being the manifold of unit octonions, and is the only one of these which does not carry (Lie) group structure but just Moufang loop structure.

Properties

Quaternionic Hopf fibration

The 7-sphere participates in the quaternionic Hopf fibration, the analog of the complex Hopf fibration with the field of complex numbers replaced by the division ring of quaternions or Hamiltonian numbers \mathbb{H}.

S 3 S 7 p S 4 BSU(2) \array{ S^3 &\hookrightarrow& S^7 \\ && \downarrow^\mathrlap{p} \\ && S^4 &\stackrel{}{\longrightarrow}& \mathbf{B} SU(2) }

Here the idea is that S 7S^7 can be construed as {(x,y) 2:|x| 2+|y| 2=1}\{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\}, with pp mapping (x,y)(x, y) to x/yx/y as an element in the projective line 1()S 4\mathbb{P}^1(\mathbb{H}) \cong S^4, with each fiber a torsor parametrized by quaternionic scalars λ\lambda of unit norm (so λS 3\lambda \in S^3). This canonical S 3S^3-bundle (or SU(2)SU(2)-bundle) is classified by a map S 4BSU(2)S^4 \to \mathbf{B} SU(2).

Smooth structures

A celebrated result of Milnor is that S 7S^7 admits exotic smooth structures (exotic spheres), i.e., there are smooth manifold structures on the topological manifold S 7S^7 that are not diffeomorphic to the standard smooth structure on S 7S^7. More structurally, considering smooth structures up to oriented diffeomorphism, the different smooth structures form a monoid under a (suitable) operation of connected sum, and this monoid is isomorphic to the cyclic group /(28)\mathbb{Z}/(28). With the notable possible exception of n=4n = 4 (where the question of existence of exotic 4-spheres is wide open), exotic spheres first occur in dimension 77. This phenomenon is connected to the h-cobordism theorem (the monoid of smooth structures is identified with the monoid of h-cobordism classes of oriented homotopy spheres?).

One explicit construction of the smooth structures is given as follows (see Milnor 1968). Let W kW_k be the algebraic variety in 5\mathbb{C}^5 defined by the equation

z 1 6k1+z 2 3+z 3 2+z 4 2+z 5 2=0z_1^{6 k - 1} + z_2^3 + z_3^2 + z_4^2 + z_5^2 = 0

and S ϵ 5S_\epsilon \subset \mathbb{C}^5 a sphere of small radius ϵ\epsilon centered at the origin. Then each of the 2828 smooth structures on S 7S^7 is represented by an intersection W kS ϵW_k \cap S_\epsilon, as kk ranges from 11 to 2828. These manifolds sometimes go by the name Brieskorn manifolds or Brieskorn sphere?s1.

G 2G_2-structure

Let ϕ 0Ω 3( 7)\phi_0 \in \Omega^3(\mathbb{R}^7) be the associative 3-form and let

Φ 0Ω 4( 7) \Phi_0 \in \Omega^4(\mathbb{R} \oplus \mathbb{R}^7)

be given by

Φ 0=dx 0ϕ 0+ϕ 0 \Phi_0 = d x_0 \wedge \phi_0 + \star \phi_0

(where x 0x_0 denotes the canonical coordinate on the first factor of \mathbb{R} and ϕ 0\phi_0 is pulled back along the projection to 7\mathbb{R}^7) .

By construction this is its own Hodge dual

Φ=Φ. \Phi = \star \Phi \,.

This implies that as we restrict Φ 0\Phi_0 to

8{0}×S 7 \mathbb{R}^8 - \{0\} \simeq \mathbb{R} \times S^7

then there is a unique 3-form

ϕΩ 3(S 7) \phi \in \Omega^3(S^7)

on the 7-sphere such that

Φ 0=r 3ϕ+r 4 S 7ϕ(on 8{0}). \Phi_0 = r^3 \wedge \phi + r^4 \star_{S^7} \phi \;\;\;\; (on \; \mathbb{R}^8 - \{0\}) \,.

This 3-form ϕ\phi defines a G2-structure on S 7S^7. It is nearly parallel in that

dϕ=4ϕ. d \phi = 4 \star \phi \,.

(e.g. Lotay 12, def.2.4)

References

  • Martin Cederwall, Christian R. Preitschopf, The Seven-sphere and its Kac-Moody Algebra, Commun. Math. Phys. 167 (1995) 373-394 (arXiv:hep-th/9309030)

  • Takeshi Ôno, On the Hopf fibration S 7S 4S^7 \to S^4 over ZZ, Nagoya Math. J. Volume 59 (1975), 59-64. (Euclid)

Relation to the Milnor fibration:

An ADE classification of finite subgroups of SO(8)SO(8) acting freely on S 7S^7 (see at group action on an n-sphere) such that the quotient is spin and has at least four Killing spinors (see also at ABJM model) is in

Discussion of exotic smooth structures on 7-spheres includes

  • Wikipedia, Exotic sphere, link.

The explicit construction of exotic 7-spheres by intersecting algebraic varieties with spheres is described in

  • John Milnor, “Singular points of complex hypersurfaces” , Princeton Univ. Press (1968).

Discussion of (nearly) G2-structures on S 7S^7 and calibrated submanifolds includes


  1. Not surprisingly, these exotic spheres are also called Milnor spheres.

Last revised on July 28, 2018 at 09:23:03. See the history of this page for a list of all contributions to it.