Contents

# Contents

## Idea

The sphere of dimension 4.

## Properties

### Coset space structure

As any sphere, the 4-sphere has the coset space structure

$S^4 \simeq O(5)/O(4) \simeq SO(5)/SO(4) \simeq Spin(5)/Spin(4)\simeq Pin(5)/Pin(4).$

There is also this:

###### Example

The coset space of Sp(2).Sp(1) (this Def.) by Sp(1)Sp(1)Sp(1) (this Def.) is the 4-sphere:

$\frac{ Sp(2)\cdot Sp(1) } { Sp(1)Sp(1)Sp(1) } \;\simeq\; S^4 \,.$

This follows essentially from the quaternionic Hopf fibration and its $Sp(2)$-equivariance

### Homotopy groups

The homotopy groups of the 4-sphere in low degree are

$k$0123456789101112
$\pi_k(S^4)$$\ast$000$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$$\mathbb{Z} \times \mathbb{Z}_2$$\mathbb{Z}_2^2$$\mathbb{Z}_2^2$$\mathbb{Z}_{24} \times \mathbb{Z}_3$$\mathbb{Z}_{15}$$\mathbb{Z}_2$

### As part of the quaternionic Hopf fibration

The 4-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 $\mathbb{H}$.

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

Here the idea is that $S^7$ may be construed as

$\array{ S^7 &\simeq S(\mathbb{H}^4) \\ & \simeq \{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\}, }$

with $p$ mapping $(x, y)$ to $x/y$ as an element in the projective line $\mathbb{P}^1(\mathbb{H}) \cong S^4$, with each fiber a torsor parameterized by quaternionic scalars $\lambda$ of unit norm (so $\lambda \in S^3$). This canonical $S^3$-bundle (or $SU(2)$-bundle) is classified by a map $S^4 \to \mathbf{B} SU(2)$.

There are other useful ways to parameterize the quaternionic Hopf fibration, such as the original Hopf construction, see there the section Hopf fibrations. By this parameterization $S^4$ is identified as $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$.

### As a quotient of the complex projective plane

The 4-sphere is the quotient space of the complex projective plane by the action on homogeneous coordinates of complex conjugation.

$\mathbb{C}P^2 / (-)^* \simeq S^4$
• Vladimir Arnold, Ramied covering $\mathbb{C}P^2 \to S^4$, hyperbolicity and projective topology, Siberian Math. Journal 1988, V. 29, N 5, P.36-47

• Vladimir Arnold, On disposition of ovals of real plane algebraic curves, involutions of four-dimensional manifolds and arithmetics of integer quadratic forms, Funct. Anal. and Its Appl., 1971, V. 5, N 3, P. 1-9.

• William Massey, The quotient space of the complex projective space under conjugation is a 4-sphere, Geometriae Didactica 1973

• Nicolaas Kuiper, The quotient space of ℂP(2) by complex conjugation is the 4-sphere, Mathematische Annalen, 1974

• J.A.Hillman, An explicit formula for a branched covering from $\mathbb{C}P^2$ to $S^4$ (arXiv:1705.05038)

### Exotic smooth structures

It is open whether the 4-sphere admits an exotic smooth structure. See (Freedman-Gompf-Morrison-Walker 09) for review.

### $SU(2)$ action

If we identify $\mathbb{R}^5 \simeq_{\mathbb{Q}} \mathbb{R} \oplus \mathbb{H}$ with the direct sum of the real line with the real vector space underlying the quaternions, so that

$S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$

as in the discussion of the quaternionic Hopf fibration above, then there is induced an action of the group SU(2) on the 4-sphere, by identifying

$SU(2) \simeq S(\mathbb{Q})$

and then acting by left multiplication.

#### Circle action

###### Proposition

Given an continuous action of the circle group on the topological 4-sphere, its fixed point space is of one of two types:

1. either it is the 0-sphere $S^0 \hookrightarrow S^4$

2. or it has the rational homotopy type of an even-dimensional sphere.

For more see at group actions on spheres.

As a special case of the $SU(2)$-action from above, we discuss the induced circle action via the embedding

$S^1 \simeq U(1) \hookrightarrow SU(2) \,.$

Consider the following circle group action on the 4-sphere:

###### Definition

($SU(2)$-action on 4-sphere)

Regard

$S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$

as the unit sphere inside the direct sum (as real vector spaces) of the real numbers with the quaternions, and regard the special unitary group $SU(2)$ as the group of unit-norm quaternions

$SU(2) \simeq S(\mathbb{H},\cdot)$

In particular this restricts to an action of the circle group

$S^1 \simeq U(1) \hookrightarrow SU(2)$

(as the diagonal matrices inside $SU(2)$) on the 4-sphere.

The resulting ordinary quotient is $S^4/_{ord} S^1 \simeq S^3$ and the projection $S^4 \to S^3$ is the suspension of the complex Hopf fibration $S^3 \to S^2$.

The fixed point set of the action is the two poles

$S^0 \;=\; \{(\pm 1, 0,0,0,0)\} \;\in\; \mathbb{R} \oplus \mathbb{H}$

introduced by the suspension, hence forms the 0-sphere space. Since this is not the empty set, the homotopy quotient $S^4 // S^1$ of the circle action differs from $S^3$, but there is still the canonical projection

$S^4//S^1 \longrightarrow S^4 / S^1 \simeq S^3 \,.$

Hence both $S^4$ and $S^4 // S^1$ are canonically homotopy types over $S^3$.

A minimal dg-module presentation in rational homotopy theory for these projections is given in Roig & Saralegi-Aranguren 00, second page:

###### Proposition

(Roig & Saralegi-Aranguren 00, p. 2)

Write

$CE(\mathfrak{l}(S^3))) = Sym^\bullet \langle \underset{\text{deg 3}}{\underbrace{h_3}} \rangle$

for the minimal Sullivan model of the 3-sphere. Then rational minimal dg-modules for the maps (via Def. )

$\array{ S^4 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^4//S^1 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^0 \\ \downarrow \\ S^3 }$

as dg-modules over $CE(\mathfrak{l}(S^3))$ are given as follows, respectively:

(1)\array{ \text{fibration} & \array{\text{vector space underlying} \\ \text{minimal dg-model}} & \array{ \text{differential on} \\ \text{minimal dg-model} } \\ \array{ S^4 \\ \downarrow \\ S^3 } & Sym^\bullet \langle \underset{ deg = 3 }{ \underbrace{ h_3 }} \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \tilde\omega_{2p} }}, \underset{deg = 2p + 4}{ \underbrace{ \omega_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \tilde\omega_0 & \mapsto 0 \\ \tilde\omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p} \\ \omega_4 & \mapsto 0 \\ \omega_{2p+6} & \mapsto h_3 \wedge \omega_{2p + 4} \end{aligned} \right. \\ \array{ S^0 \\ \downarrow \\ S^3 } & Sym^\bullet \langle \underset{ deg = 3 }{ \underbrace{ h_3 }} \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \tilde \omega_{2p} }}, \underset{ deg = 2p }{ \underbrace{ \omega_{2p} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \tilde \omega_0 & \mapsto 0 \\ \tilde \omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p} \\ \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \end{aligned} \right. \\ \array{ S^4//S^1 \\ \downarrow \\ S^3 } & Sym^\bullet \langle \underset{ deg = 3 }{ \underbrace{ h_3 }} , \underset{ deg = 2 }{ \underbrace{ \omega_2 }} \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \tilde \omega_{2p} }}, \underset{ deg =2p + 4 }{ \underbrace{ \omega_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \tilde \omega_0 & \mapsto 0 \\ \tilde \omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p} \\ \omega_2 & \mapsto 0 \\ \omega_{2p+4} & \mapsto h_3 \wedge \omega_{2p + 2} \end{aligned} \right. }

Beware that in the model for $S^4//S^2$ the element $\omega_2$ induces its entire polynomial algebra as generator of the dg-module.

Notice that we changed the notation of the generators compared to Roig & Saralegi-Aranguren 00, second page, to bring out the pattern:

$\phantom{A}$Roig$\phantom{A}$$\phantom{A}$here$\phantom{A}$
$\phantom{A}a\phantom{A}$$\phantom{A}h_3\phantom{A}$
$\phantom{A}1\phantom{A}$$\phantom{A}\tilde\omega_0\phantom{A}$
$\phantom{A}c_{2n}\phantom{A}$$\phantom{A}\tilde\omega_{2n+2}\phantom{A}$
$\phantom{A}c_{2n+1}\phantom{A}$$\phantom{A}\omega_{2n+4}\phantom{A}$
$\phantom{A}e\phantom{A}$$\phantom{A}\omega_2\phantom{A}$
$\phantom{A}\gamma_{2n}\phantom{A}$$\phantom{A}\tilde\omega_{2n}\phantom{A}$
$\phantom{A}\gamma_{2n+1}\phantom{A}$$\phantom{A}\omega_{2n}\phantom{A}$

#### M5-brane orbifolds

The supersymmetric Freund-Rubin compactifications of 11-dimensional supergravity which are Cartesian products of 7-dimensional anti-de Sitter spacetime with a compact 4-dimensional orbifold

$AdS_7 \times X_4$

(the near horizon geometry of a black M5-brane) are all of the form

$X_4 \simeq S^4//G$

where $G \subset SU(2)$ is a finite subgroup of $SU(2)$ (i.e. an ADE group), acting via the identification $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$ as above, and where the double slash denotes the homotopy quotient (orbifold quotient).

### Free and cyclic loop space

We discuss the rational homotopy theory of the free loop space $\mathcal{L}(S^4)$ of $S^4$, as well as the cyclic loop space $\mathcal{L}(S^4)/S^1$ using the results from Sullivan models of free loop spaces:

###### Example

Let $X = S^4$ be the 4-sphere. The corresponding rational n-sphere has minimal Sullivan model

$(\wedge^\bullet \langle g_4, g_7 \rangle, d)$

with

$d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4 \,.$

Hence this prop. gives for the rationalization of $\mathcal{L}S^4$ the model

$( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4} )$

with

\begin{aligned} d_{\mathcal{L}S^4} h_3 & = 0 \\ d_{\mathcal{L}S^4} \omega_4 & = 0 \\ d_{\mathcal{L}S^4} \omega_6 & = h_3 \wedge \omega_4 \\ d_{\mathcal{L}S^4} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 \\ \end{aligned}

and this prop gives for the rationalization of $\mathcal{L}S^4 / / S^1$ the model

$( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4 / / S^1} )$

with

\begin{aligned} d_{\mathcal{L}S^4 / / S^1} h_3 & = 0 \\ d_{\mathcal{L}S^4 / / S^1} \omega_2 & = 0 \\ d_{\mathcal{L}S^4 / / S^1} \omega_4 & = h_3 \wedge \omega_2 \\ d_{\mathcal{L}S^4 / / S^1} \omega_6 & = h_3 \wedge \omega_4 \\ d_{\mathcal{L}S^4 / / S^1} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 + \omega_2 \wedge \omega_6 \end{aligned} \,.
###### Proposition

Let $\hat \mathfrak{g} \to \mathfrak{g}$ be a central Lie algebra extension by $\mathbb{R}$ of a finite dimensional Lie algebra $\mathfrak{g}$, and let $\mathfrak{g} \longrightarrow b \mathbb{R}$ be the corresponding L-∞ 2-cocycle with coefficients in the line Lie 2-algebra $b \mathbb{R}$, hence (FSS 13, prop. 3.5) so that there is a homotopy fiber sequence of L-∞ algebras

$\hat \mathfrak{g} \longrightarrow \mathfrak{g} \overset{\omega_2}{\longrightarrow} b \mathbb{R}$

which is dually modeled by

$CE(\hat \mathfrak{g}) = ( \wedge^\bullet ( \mathfrak{g}^\ast \oplus \langle e \rangle ), d_{\hat \mathfrak{g}}|_{\mathfrak{g}^\ast} = d_{\mathfrak{g}},\; d_{\hat \mathfrak{g}} e = \omega_2) \,.$

For $X$ a space with Sullivan model $(A_X,d_X)$ write $\mathfrak{l}(X)$ for the corresponding L-∞ algebra, i.e. for the $L_\infty$-algebra whose Chevalley-Eilenberg algebra is $(A_X,d_X)$:

$CE(\mathfrak{l}X) = (A_X,d_X) \,.$

Then there is an isomorphism of hom-sets

$Hom_{L_\infty Alg}( \hat \mathfrak{g}, \mathfrak{l}(S^4) ) \;\simeq\; Hom_{L_\infty Alg/b \mathbb{R}}( \mathfrak{g}, \mathfrak{l}( \mathcal{L}S^4 / S^1 ) ) \,,$

with $\mathfrak{l}(S^4)$ from this prop. and $\mathfrak{l}(\mathcal{L}S^4 //S^1)$ from this prop., where on the right we have homs in the slice over the line Lie 2-algebra, via this prop.

Moreover, this isomorphism takes

$\hat \mathfrak{g} \overset{(g_4, g_7)}{\longrightarrow} \mathfrak{l}(S^4)$

to

$\array{ \mathfrak{g} && \overset{(\omega_2,\omega_4, \omega_6, h_3,h_7)}{\longrightarrow} && \mathfrak{l}( \mathcal{L}X / S^1 ) \\ & {}_{\mathllap{\omega_2}}\searrow && \swarrow_{\mathrlap{\omega_2}} \\ && b \mathbb{R} } \,,$

where

$\omega_4 = g_4 - h_3 \wedge e \;\,, \;\;\; h_7 = g_7 + \omega_6 \wedge e$

with $e$ being the central generator in $CE(\hat \mathfrak{g})$ from above, and where the equations take place in $\wedge^\bullet \hat \mathfrak{g}^\ast$ with the defining inclusion $\wedge^\bullet \mathfrak{g}^\ast \hookrightarrow \wedge^\bullet \mathfrak{g}^\ast$ understood.

This is observed in (FSS 16, FSS 16b), where it serves to formalize, on the level of rational homotopy theory, the double dimensional reduction of M-branes in M-theory to D-branes in type IIA string theory (for the case that $\mathfrak{g}$ is type IIA super Minkowski spacetime $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ and $\hat \mathfrak{g}$ is 11d super Minkowski spacetime $\mathbb{R}^{10,1\vert \mathbf{32}}$, and the cocycles are those of The brane bouquet).

###### Proof

By the fact that the underlying graded algebras are free, and since $e$ is a generator of odd degree, the given decomposition for $\omega_4$ and $h_7$ is unique.

Hence it is sufficient to observe that under this decomposition the defining equations

$d g_4 = 0 \,,\;\;\; d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4$

for the $\mathfrak{l}S^4$-valued cocycle on $\hat \mathfrak{g}$ turn into the equations for a $\mathfrak{l} ( \mathcal{L}S^4 / S^1 )$-valued cocycle on $\mathfrak{g}$. This is straightforward:

\begin{aligned} & d_{\hat \mathfrak{g}} ( \omega_4 + h_3 \wedge e ) = 0 \\ \Leftrightarrow \;\;\;\; & d_{\mathfrak{g}} (\omega_4 - h_3 \wedge \omega_2) = 0 \;\;\; and \;\;\; d_{\mathfrak{g}} h_3 = 0 \\ \Leftrightarrow \;\;\;\; & d_{\mathfrak{g}} \omega_4 = h_3 \wedge \omega_2 \;\;\; and \;\;\; d_{\mathfrak{g}} h_3 = 0 \end{aligned}

as well as

\begin{aligned} & d_{\hat \mathfrak{g}} ( h_7 - \omega_6 \wedge e ) = -\tfrac{1}{2}( \omega_4 + h_3 \wedge e ) \wedge (\omega_4 + h_3\wedge e) \\ \Leftrightarrow \;\;\;\; & d_\mathfrak{g} h_7 - \omega_6 \wedge \omega_2 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 \;\;\; and \;\;\; - d_\mathfrak{g} \omega_6 = - h_3 \wedge \omega_4 \\ \Leftrightarrow \;\;\;\; & d_\mathfrak{g} h_7 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 + \omega_6 \wedge \omega_2 \;\;\; and \;\;\; d_\mathfrak{g} h_6 = h_3 \wedge \omega_4 \end{aligned}
$\infty Grpd \underoverset {\underset{\mathcal{L}(-)/S^1}{\longrightarrow}} {\overset{hofib}{\longleftarrow}} {\bot} \infty Grpd_{/B S^1}$

(this prop.) applied to the $S^1$-principal infinity-bundle

$\array{ S^4 \\ {}^{\mathllap{hofib(c)}}\downarrow \\ S^4 / S^1 &\underset{c}{\longrightarrow}& B S^1 }$

is a natural map

$S^4/S^1 \longrightarrow \mathcal{L}(S^4)/S^1$

from the homotopy quotient by the circle action (def. ), to the cyclic loop space of the 4-sphere.

spheres – contents

### Branched covers

All PL 4-manifolds are simple branched covers of the 4-sphere:

Speculative remarks on the possible role of maps from spacetime to the 4-sphere in some kind of quantum gravity via spectral geometry (related to the Connes-Lott-Chamseddine-Barrett model) are in