nLab 4-sphere

Idea

The sphere of dimension 4.

Properties

Basic differential geometry

An embedded radius-$R$ 4-sphere inherits a volume form (a degree-4 differential form from the ambient $5$-dimensional Euclidean space, namely

$$\omega = \frac{1}{R} \sum_{j=1}^{5} (-1)^{j-1} x_j d x_1 \wedge \ldots \wedge \widehat{d x_j} \wedge \ldots \wedge d x_{5}$$

where the hat means omit that factor. This is equal to $\ast d r$, where $\ast$ is the Hodge star operator in $\mathbb{R}^5$ for the Euclidean metric, and $d r$ is the exterior derivative of the radius function.

The volume of the manifold $S^4$ with this volume form is then given by $8\pi^2R^4/3$.

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…

(e.g. Bettiol-Mendes 15, (3.1), (3.2), (3.3))

Homotopy groups

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

$k$	0	1	2	3	4	5	6	7	8	9	10	11	12
$\pi_k(S^4)$	$\ast$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z} \times \mathbb{Z}_{12}$	$\mathbb{Z}_2^2$	$\mathbb{Z}_2^2$	$\mathbb{Z}_{24} \times \mathbb{Z}_3$	$\mathbb{Z}_{15}$	$\mathbb{Z}_2$

For more see at Serre finiteness theorem and at homotopy groups of spheres.

Bundles over the 4-sphere

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

The Calabi-Penrose fibration

See at Calabi-Penrose fibration.

The complex projective plane

Proposition

(Arnold-Kuiper-Massey theorem)

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

$$\mathbb{C}P^2 / (-)^* \simeq S^4$$

Exotic smooth structures

It is open whether the 4-sphere admits an exotic smooth structure. See Freedman, Gompf, Morrison & Walker 2009 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.

(Félix-Oprea-Tanré 08, Example 7.39)

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).

See (AFHS 98, section 5.2, MF 12, section 8.3).

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

$$\begin{aligned} & d_{\hat \mathfrak{g}} ( h_7 - \omega_6 \wedge e ) = -\tfrac{1}{2}( \omega_4 + h_3 \wedge e ) \wedge (\omega_4 + h_3\wedge e) \\ \Leftrightarrow \;\;\;\; & d_\mathfrak{g} h_7 - \omega_6 \wedge \omega_2 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 \;\;\; and \;\;\; - d_\mathfrak{g} \omega_6 = - h_3 \wedge \omega_4 \\ \Leftrightarrow \;\;\;\; & d_\mathfrak{g} h_7 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 + \omega_6 \wedge \omega_2 \;\;\; and \;\;\; d_\mathfrak{g} h_6 = h_3 \wedge \omega_4 \end{aligned}$$

The unit of the double dimensional reduction-adjunction

$$\infty Grpd \underoverset {\underset{\mathcal{L}(-)/S^1}{\longrightarrow}} {\overset{hofib}{\longleftarrow}} {\bot} \infty Grpd_{/B S^1}$$

(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.

Diffeomorphism group

Counterexamples (via graph complexes) to the analogue of the Smale conjecture for the 4-sphere are claimed in Watanabe 18, reviewed in Watanabe 21.

Related entries

spheres – contents

• fuzzy 4-sphere

• 2-sphere

• 3-sphere

• 5-sphere

• 6-sphere

• 7-sphere

• n-sphere

References

General

• Michael Freedman, Robert Gompf, Scott Morrison, Kevin Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, Quantum Topology, Volume 1, Issue 2 (2010), pp. 171-208 (arXiv:0906.5177)

• Agustí Roig, Martintxo Saralegi-Aranguren, Minimal Models for Non-Free Circle Actions, Illinois Journal of Mathematics, volume 44, number 4 (2000) (arXiv:math/0004141)

• Bobby Acharya, José Figueroa-O'Farrill, Chris Hull, B. Spence, Branes at conical singularities and holography , Adv. Theor. Math. Phys. 2 (1998) 1249–1286

• Yves Félix, John Oprea, Daniel Tanré, Algebraic Models in Geometry, Oxford University Press 2008

• Paul de Medeiros, José Figueroa-O'Farrill, Half-BPS M2-brane orbifolds, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (arXiv:1007.4761, Euclid)

• Renato G. Bettiol, Ricardo A. E. Mendes, Flag manifolds with strongly positive curvature, Math. Z. 280 (2015), no. 3-4, 1031-1046 (arXiv:1412.0039)

• Selman Akbulut, Homotopy 4-spheres associated to an infinite order loose cork (arXiv:1901.08299)

• Akio Kawauchi, Smooth homotopy 4-sphere (arXiv:1911.11904)

• David T. Gay, Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins (arXiv:2102.12890)

Branched covers

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

• Riccardo Piergallini, Four-manifolds as 4-fold branched covers of $S^4$, Topology Volume 34, Issue 3, July 1995 (doi:10.1016/0040-9383(94)00034-I, pdf)

• Massimiliano Iori, Riccardo Piergallini, 4-manifolds as covers of the 4-sphere branched over non-singular surfaces, Geom. Topol. 6 (2002) 393-401 (arXiv:math/0203087)

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

• Ali Chamseddine, Alain Connes, Viatcheslav Mukhanov, Quanta of Geometry: Noncommutative Aspects, Phys. Rev. Lett. 114 (2015) 9, 091302 (arXiv:1409.2471)

• Ali Chamseddine, Alain Connes, Viatcheslav Mukhanov, Geometry and the Quantum: Basics, JHEP 12 (2014) 098 (arXiv:1411.0977)

• Alain Connes, section 4 of Geometry and the Quantum, in Foundations of Mathematics and Physics One Century After Hilbert, Springer 2018. 159-196 (arXiv:1703.02470, doi:10.1007/978-3-319-64813-2)

• Alain Connes, from 58:00 to 1:25:00 in Why Four Dimensions and the Standard Model Coupled to Gravity - A Tentative Explanation From the New Geometric Paradigm of NCG, talk at IHES, 2017 (video recording)