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4-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 4.

Properties

Coset space structure\

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

S 4O(5)/O(4)SO(5)/SO(4)Spin(5)/Spin(4)Pin(5)/Pin(4). 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:

Sp(2)Sp(1)Sp(1)Sp(1)Sp(1)S 4. \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)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

kk0123456789101112
π k(S 4)\pi_k(S^4)*\ast000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2× 2\mathbb{Z} \times \mathbb{Z}_2 2 2\mathbb{Z}_2^2 2 2\mathbb{Z}_2^2 24× 3\mathbb{Z}_{24} \times \mathbb{Z}_3 15\mathbb{Z}_{15} 2\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}.

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 may be construed as

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

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 4S^4 is identified as S 4S()S^4 \simeq S(\mathbb{R} \oplus \mathbb{H}).

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)SU(2) action

If we identify 5 \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 4S() 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)S() 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 0S 4S^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)SU(2)-action from above, we discuss the induced circle action via the embedding

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

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

Definition

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

Regard

S 4S() 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)SU(2) as the group of unit-norm quaternions

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

In particular this restricts to an action of the circle group

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

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

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

The fixed point set of the action is the two poles

S 0={(±1,0,0,0,0)} 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 1S^4 // S^1 of the circle action differs from S 3S^3, but there is still the canonical projection

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

Hence both S 4S^4 and S 4//S 1S^4 // S^1 are canonically homotopy types over S 3S^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(𝔩(S 3)))=Sym h 3deg 3 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. )

S 4 S 3,AAS 4//S 1 S 3,AAS 0 S 3 \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(𝔩(S 3))CE(\mathfrak{l}(S^3)) are given as follows, respectively:

(1)fibration vector space underlying minimal dg-model differential on minimal dg-model S 4 S 3 Sym h 3deg=3ω˜ 2pdeg=2p,ω 2p+4deg=2p+4|p d:{ω˜ 0 0 ω˜ 2p+2 h 3ω˜ 2p ω 4 0 ω 2p+6 h 3ω 2p+4 S 0 S 3 Sym h 3deg=3ω˜ 2pdeg=2p,ω 2pdeg=2p|p d:{ω˜ 0 0 ω˜ 2p+2 h 3ω˜ 2p ω 0 0 ω 2p+2 h 3ω 2p S 4//S 1 S 3 Sym h 3deg=3,ω 2deg=2ω˜ 2pdeg=2p,ω 2p+4deg=2p+4|p d:{ω˜ 0 0 ω˜ 2p+2 h 3ω˜ 2p ω 2 0 ω 2p+4 h 3ω 2p+2 \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 2S^4//S^2 the element ω 2\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:

A\phantom{A}RoigA\phantom{A}A\phantom{A}hereA\phantom{A}
AaA\phantom{A}a\phantom{A}Ah 3A\phantom{A}h_3\phantom{A}
A1A\phantom{A}1\phantom{A}Aω˜ 0A\phantom{A}\tilde\omega_0\phantom{A}
Ac 2nA\phantom{A}c_{2n}\phantom{A}Aω˜ 2n+2A\phantom{A}\tilde\omega_{2n+2}\phantom{A}
Ac 2n+1A\phantom{A}c_{2n+1}\phantom{A}Aω 2n+4A\phantom{A}\omega_{2n+4}\phantom{A}
AeA\phantom{A}e\phantom{A}Aω 2A\phantom{A}\omega_2\phantom{A}
Aγ 2nA\phantom{A}\gamma_{2n}\phantom{A}Aω˜ 2nA\phantom{A}\tilde\omega_{2n}\phantom{A}
Aγ 2n+1A\phantom{A}\gamma_{2n+1}\phantom{A}Aω 2nA\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×X 4 AdS_7 \times X_4

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

X 4S 4//G X_4 \simeq S^4//G

where GSU(2)G \subset SU(2) is a finite subgroup of SU(2)SU(2) (i.e. an ADE group), acting via the identification S 4S()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 (S 4)\mathcal{L}(S^4) of S 4S^4, as well as the cyclic loop space (S 4)/S 1\mathcal{L}(S^4)/S^1 using the results from Sullivan models of free loop spaces:

Example

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

( g 4,g 7,d) (\wedge^\bullet \langle g_4, g_7 \rangle, d)

with

dg 4=0,dg 7=12g 4g 4. d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4 \,.

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

( ω 4,ω 6,h 3,h 7,d S 4) ( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4} )

with

d S 4h 3 =0 d S 4ω 4 =0 d S 4ω 6 =h 3ω 4 d S 4h 7 =12ω 4ω 4 \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 S 4//S 1\mathcal{L}S^4 / / S^1 the model

( ω 2,ω 4,ω 6,h 3,h 7,d S 4//S 1) ( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4 / / S^1} )

with

d S 4//S 1h 3 =0 d S 4//S 1ω 2 =0 d S 4//S 1ω 4 =h 3ω 2 d S 4//S 1ω 6 =h 3ω 4 d S 4//S 1h 7 =12ω 4ω 4+ω 2ω 6. \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 𝔤^𝔤\hat \mathfrak{g} \to \mathfrak{g} be a central Lie algebra extension by \mathbb{R} of a finite dimensional Lie algebra 𝔤\mathfrak{g}, and let 𝔤b\mathfrak{g} \longrightarrow b \mathbb{R} be the corresponding L-∞ 2-cocycle with coefficients in the line Lie 2-algebra bb \mathbb{R}, hence (FSS 13, prop. 3.5) so that there is a homotopy fiber sequence of L-∞ algebras

𝔤^𝔤ω 2b \hat \mathfrak{g} \longrightarrow \mathfrak{g} \overset{\omega_2}{\longrightarrow} b \mathbb{R}

which is dually modeled by

CE(𝔤^)=( (𝔤 *e),d 𝔤^| 𝔤 *=d 𝔤,d 𝔤^e=ω 2). 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 XX a space with Sullivan model (A X,d X)(A_X,d_X) write 𝔩(X)\mathfrak{l}(X) for the corresponding L-∞ algebra, i.e. for the L L_\infty-algebra whose Chevalley-Eilenberg algebra is (A X,d X)(A_X,d_X):

CE(𝔩X)=(A X,d X). CE(\mathfrak{l}X) = (A_X,d_X) \,.

Then there is an isomorphism of hom-sets

Hom L Alg(𝔤^,𝔩(S 4))Hom L Alg/b(𝔤,𝔩(S 4/S 1)), 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 𝔩(S 4)\mathfrak{l}(S^4) from this prop. and 𝔩(S 4//S 1)\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

𝔤^(g 4,g 7)𝔩(S 4) \hat \mathfrak{g} \overset{(g_4, g_7)}{\longrightarrow} \mathfrak{l}(S^4)

to

𝔤 (ω 2,ω 4,ω 6,h 3,h 7) 𝔩(X/S 1) ω 2 ω 2 b, \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

ω 4=g 4h 3e,h 7=g 7+ω 6e \omega_4 = g_4 - h_3 \wedge e \;\,, \;\;\; h_7 = g_7 + \omega_6 \wedge e

with ee being the central generator in CE(𝔤^)CE(\hat \mathfrak{g}) from above, and where the equations take place in 𝔤^ *\wedge^\bullet \hat \mathfrak{g}^\ast with the defining inclusion 𝔤 * 𝔤 *\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 𝔤\mathfrak{g} is type IIA super Minkowski spacetime 9,1|16+16¯\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} and 𝔤^\hat \mathfrak{g} is 11d super Minkowski spacetime 10,1|32\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 ee is a generator of odd degree, the given decomposition for ω 4\omega_4 and h 7h_7 is unique.

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

dg 4=0,dg 7=12g 4g 4 d g_4 = 0 \,,\;\;\; d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4

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

d 𝔤^(ω 4+h 3e)=0 d 𝔤(ω 4h 3ω 2)=0andd 𝔤h 3=0 d 𝔤ω 4=h 3ω 2andd 𝔤h 3=0 \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

d 𝔤^(h 7ω 6e)=12(ω 4+h 3e)(ω 4+h 3e) d 𝔤h 7ω 6ω 2=12ω 4ω 4andd 𝔤ω 6=h 3ω 4 d 𝔤h 7=12ω 4ω 4+ω 6ω 2andd 𝔤h 6=h 3ω 4 \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

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

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

S 4 hofib(c) S 4/S 1 c BS 1 \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(S 4)/S 1 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.

References

General

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

Last revised on May 7, 2019 at 10:40:32. See the history of this page for a list of all contributions to it.