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real Hopf fibration

Contents

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

The real Hopf fibration is the fibration

S 0 S 1 p S 1 \array{ S^0 &\hookrightarrow& S^1 \\ && \downarrow^{\mathrlap{p_{\mathbb{R}}}} \\ && S^1 }

of the 1-sphere over itself with fiber the 0-sphere, which is induced via the Hopf construction from the product operation

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

on the real numbers.

This may also be understood as the Spin(2)-double cover of SO(2).

Realizations

Here are different but equivalent ways of realizing this explicitly:

Via projective space

If the domain S 1S^1 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{R}\times \mathbb{R} and the codomain S 1S^1 is regarded as the real projective space, then pp is given simply by

p:(x,y)[x;y]=[x/y;1], p \colon (x,y) \mapsto [x;y] = [x/y; 1] \,,

One can view the real Hopf fibration as the boundary of a Möbius strip, which is the non-trivial double cover of the circle.

As an element in the first (stable) homotopy group of spheres π 1(S 1)\pi_1(S^1) \simeq \mathbb{Z}, the real Hopf fibration represents p =2p_{\mathbb{R}} = 2.

Via join construction

We spell out the real Hopf fibration realized as the Hopf construction

X*X H ΣX = = (X×[0,1]×X) / (X×[0,1]) / (x,t,y) (f(x,y),t) \array{ X \ast X &\overset{H_{\mathbb{R}}}{\longrightarrow}& \Sigma X \\ = && = \\ (X \times [0,1] \times X)_{/\sim} && (X \times [0,1])_{/\sim} \\ (x,t,y) &\mapsto& \big( f(x,y), t\big) }

(as defined there) on

X/2 X \coloneqq \mathbb{Z}/2

regarded with its cyclic group-structure

(/2)×(/2) f (/2) (x,y) x+y \array{ (\mathbb{Z}/2) \times (\mathbb{Z}/2) &\overset{f}{\longrightarrow}& (\mathbb{Z}/2) \\ (x,y) &\mapsto& x + y }

Here the equivalence relation for the suspension ΣX=(X×[0,1]) /\Sigma X = (X \times [0,1])_{/\sim} on the right is

(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

while the equivalence relation for the join X*X=(X×[0,1]×X) /X\ast X = (X \times [0,1] \times X)_{/\sim} on the left is

(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) \,.

So on the left we have a circle S 1S^1 realized by gluing four copies of the interval [0,1][0,1] labeled in /2×/2\mathbb{Z}/2 \times \mathbb{Z}/2 by identifying them pairwise at 0[0,1]0 \in [0,1] and pairwise the other way at 1[0,1]1 \in [0,1]. while on the right we have a circle S 1S^1 realized by two copies, labeled by /2\mathbb{Z}/2.

S 1S 1. S^1 \longrightarrow S^1 \,.

A full path around the circle on the left is given, in terms of the above coordinates in [0,1]×( 2× 2)[0,1] \times (\mathbb{Z}_2 \times \mathbb{Z}_2), by

(0,(0,0)) (1,(0,0)) (1,(1,0)) (0,(1,0)) (0,(1,1)) (1,(1,1)) (1,(0,1)) (0,(0,1)) (0,(0,0)) \array{ (0,(0,0)) &\rightsquigarrow& (1,(0,0)) \\ && \sim \\ && (1,(1,0)) &\rightsquigarrow& (0,(1,0)) \\ &&&& \sim \\ &&&& (0,(1,1)) &\rightsquigarrow& (1,(1,1)) \\ &&&&&& \sim \\ &&&&&& (1,(0,1)) & \rightsquigarrow& (0,(0,1)) \\ &&&&&&&& \sim \\ &&&&&&&& (0,(0,0)) }

As we map this path over to the other S 1S^1 by adding up the two coordinate labels in /2\mathbb{Z}/2, we trace out the following path on the right, with coordinates in [0,1]×/2[0,1] \times \mathbb{Z}/2:

(0,(0+0=0)) (1,(0+0=0)) (1,(1+0=1)) (0,(1+0=1)) (0,(1+1=0)) (1,(1+1=0)) (1,(0+1=1)) (0,(0+1=1)) (0,(0+0=0)) \array{ (0,(0+0 = 0)) &\rightsquigarrow& (1,(0 + 0 = 0)) \\ && \sim \\ && (1,(1 + 0 = 1)) &\rightsquigarrow& (0,(1 + 0 = 1)) \\ && && \sim \\ && && (0,(1+1 = 0)) &\rightsquigarrow& (1,(1+1 = 0)) \\ && && && \sim \\ && && && (1,(0 + 1 = 1)) &\rightsquigarrow& (0,(0 + 1 = 1)) \\ && && && && \sim \\ && && && && (0,(0 + 0 = 0)) }

That’s twice around the circle on the right, for once on the left, manifestly showing that the real Hopf fibration is the non-trivial double cover of the circle by itself:

S 12S 1. S^1 \overset{\cdot 2}{\longrightarrow} S^1 \,.

Last revised on February 18, 2019 at 10:09:05. See the history of this page for a list of all contributions to it.