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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> 4-sphere in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } </style> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script type="text/javascript"> <![CDATA[//><!]]> </script> </head> <body> <div id="Container"> <textarea id='content' readonly=' readonly' rows='24' cols='60' > +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Spheres +--{: .hide} [[!include spheres -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The [[sphere]] of [[dimension]] 4. ## Properties ### Basic differential geometry {#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 [[Riemannian metric|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 {#CosetSpaceStructure} 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: +-- {: .num_example #Sp2Sp1BySp1Sp1Sp1IsS4} ###### Example The [[coset space]] of [[Sp(2).Sp(1)]] ([this Def.](SpnSp1#SpnSp1)) by [[Sp(1)Sp(1)Sp(1)]] ([this Def.](SpnSp1#Spin4Spin3)) 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 $Sp(2)$-[[equivariant function|equivariance]]... =-- (e.g. [Bettiol-Mendes 15, (3.1), (3.2), (3.3)](#BettiolMendes15)) ### Homotopy groups {#HomotopyGroups} 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 $S^4 \to \mathbf{B} SU(2)$. {#HopfParameterization} There are other useful ways to parameterize the quaternionic Hopf fibration, such as the original _[[Hopf construction]]_, see there the section _[Hopf fibrations](Hopf+construction#HopfFibrations)_. 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 {#AsAQuotientOfTheComplexProjectivePlane} +-- {: .num_prop} ###### 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](#FreedmanGompfMorrisonWalker09) for review. ### $SU(2)$ action {#QuaternionAction} 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](#HopfParameterization), then there is induced an [[action]] of the group [[special unitary group|SU(2)]] on the 4-sphere, by identifying $$ SU(2) \simeq S(\mathbb{Q}) $$ and then acting by left multiplication. #### Circle action {#CircleAction} +-- {: .num_prop} ###### Proposition Given an continuous [[action]] of the [[circle group]] on the [[topological space|topological]] [[4-sphere]], its [[fixed point]] space is of one of two types: 1. either it is the [[0-sphere]] $S^0 \hookrightarrow S^4$ 1. or it has the [[rational homotopy theory|rational homotopy type]] of an even-dimensional sphere. =-- ([Félix-Oprea-Tanré 08, Example 7.39](#FelixOpreaTanre08)) For more see at _[[group actions on spheres]]_. As a special case of the $SU(2)$-action from [above](#QuaternionAction), we discuss the induced circle action via the embedding $$ S^1 \simeq U(1) \hookrightarrow SU(2) \,. $$ Consider the following [[circle group|circle]] [[group action on an n-sphere|group action on the 4-sphere]]: +-- {: .num_defn #CircleActionOn4Sphere} ###### 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](#RoigSaralegiAranguren00): +-- {: .num_prop #FourSphereOverThreeSphereMinimalDgModel} ###### Proposition **([Roig & Saralegi-Aranguren 00, p. 2](#RoigSaralegiAranguren00))** 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 homotopy theory|rational]] [[minimal dg-modules]] for the maps (via Def. \ref{CircleActionOn4Sphere}) $$ \array{ S^4 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^4//S^1 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^0 \\ \downarrow \\ S^3 } $$ as [[dg-modules]] over $CE(\mathfrak{l}(S^3))$ are given as follows, respectively: $$ \label{FourSphereAndRelatedOverThreeSphereMinimalDGModels} \array{ \text{fibration} & \array{\text{vector space underlying} \\ \text{minimal dg-model}} & \array{ \text{differential on} \\ \text{minimal dg-model} } \\ \array{ S^4 \\ \downarrow \\ S^3 } & Sym^\bullet \langle \underset{ deg = 3 }{ \underbrace{ h_3 }} \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \tilde\omega_{2p} }}, \underset{deg = 2p + 4}{ \underbrace{ \omega_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \tilde\omega_0 & \mapsto 0 \\ \tilde\omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p} \\ \omega_4 & \mapsto 0 \\ \omega_{2p+6} & \mapsto h_3 \wedge \omega_{2p + 4} \end{aligned} \right. \\ \array{ S^0 \\ \downarrow \\ S^3 } & Sym^\bullet \langle \underset{ deg = 3 }{ \underbrace{ h_3 }} \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \tilde \omega_{2p} }}, \underset{ deg = 2p }{ \underbrace{ \omega_{2p} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \tilde \omega_0 & \mapsto 0 \\ \tilde \omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p} \\ \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \end{aligned} \right. \\ \array{ S^4//S^1 \\ \downarrow \\ S^3 } & Sym^\bullet \langle \underset{ deg = 3 }{ \underbrace{ h_3 }} , \underset{ deg = 2 }{ \underbrace{ \omega_2 }} \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \tilde \omega_{2p} }}, \underset{ deg =2p + 4 }{ \underbrace{ \omega_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \tilde \omega_0 & \mapsto 0 \\ \tilde \omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p} \\ \omega_2 & \mapsto 0 \\ \omega_{2p+4} & \mapsto h_3 \wedge \omega_{2p + 2} \end{aligned} \right. } $$ =-- 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](#RoigSaralegiAranguren00), 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 [[supersymmetry|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 brane|black]] [[M5-brane]]) are all of the form $$ X_4 \simeq S^4//G $$ where $G \subset SU(2)$ is a [[finite group|finite]] [[subgroup]] of $SU(2)$ (i.e. an [[ADE classification|ADE group]]), [[action|acting]] via the identification $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$ as [above](#QuaternionAction), and where the double slash denotes the [[homotopy quotient]] ([[orbifold quotient]]). See ([AFHS 98, section 5.2](#AFHS98), [MF 12, section 8.3](#MF12)). ### Free and cyclic loop space {#FreeLoopSpace} 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]]_: +-- {: .num_example} ###### 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.](Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace) 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](Sullivan+model+of+free+loop+space#ModelForS1quotient) 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 $$ \begin{aligned} d_{\mathcal{L}S^4 / / S^1} h_3 & = 0 \\ d_{\mathcal{L}S^4 / / S^1} \omega_2 & = 0 \\ d_{\mathcal{L}S^4 / / S^1} \omega_4 & = h_3 \wedge \omega_2 \\ d_{\mathcal{L}S^4 / / S^1} \omega_6 & = h_3 \wedge \omega_4 \\ d_{\mathcal{L}S^4 / / S^1} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 + \omega_2 \wedge \omega_6 \end{aligned} \,. $$ =-- +-- {: .num_prop} ###### Proposition Let $\hat \mathfrak{g} \to \mathfrak{g}$ be a [[central extension|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-∞ algebra cohomology|L-∞ 2-cocycle]] with coefficients in the [[line Lie n-algebra|line Lie 2-algebra]] $b \mathbb{R}$, hence ([[schreiber:The brane bouquet|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.](Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace) and $\mathfrak{l}(\mathcal{L}S^4 //S^1)$ from [this prop.](Sullivan+model+of+free+loop+space#ModelForS1quotient), where on the right we have homs in the [[slice category|slice]] over the [[line Lie n-algebra|line Lie 2-algebra]], via [this prop.](Sullivan+model+of+free+loop+space#ModelForS1quotient) 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](Sullivan+model+of+free+loop+space#FiorenzaSatiSchreiber16), [FSS 16b](#FSS16b)), 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 [[schreiber:The brane bouquet]]). +-- {: .proof} ###### 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 an adjunction|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.](double+dimensional+reduction#GeneralReduction)) 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. \ref{CircleActionOn4Sphere}), 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](diffeomorphism+group#Watanabe18), reviewed in [Watanabe 21](diffeomorphism+group#Watanabe21). ## Related entries [[spheres -- contents]] * [[fuzzy 4-sphere]] * [[2-sphere]] * [[3-sphere]] * [[5-sphere]] * [[6-sphere]] * [[7-sphere]] * [[n-sphere]] ## References ### General * {#FreedmanGompfMorrisonWalker09} [[Michael Freedman]], [[Robert Gompf]], [[Scott Morrison]], [[Kevin Walker]], _Man and machine thinking about the smooth 4-dimensional Poincar&#233; conjecture_, Quantum Topology, Volume 1, Issue 2 (2010), pp. 171-208 ([arXiv:0906.5177](http://arxiv.org/abs/0906.5177)) * {#RoigSaralegiAranguren00} [[Agustí Roig]], [[Martintxo Saralegi-Aranguren]], _Minimal Models for Non-Free Circle Actions_, Illinois Journal of Mathematics, volume 44, number 4 (2000) ([arXiv:math/0004141](https://arxiv.org/abs/math/0004141)) * {#AFHS98} [[Bobby Acharya]], [[José Figueroa-O'Farrill]], [[Chris Hull]], B. Spence, _Branes at conical singularities and holography_ , Adv. Theor. Math. Phys. 2 (1998) 1249&#8211;1286 * {#FelixOpreaTanre08} [[Yves Félix]], John Oprea, [[Daniel Tanré]], _Algebraic Models in Geometry_, Oxford University Press 2008 * {#MF12} [[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](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553)) * {#BettiolMendes15} 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](https://arxiv.org/abs/1412.0039)) * Selman Akbulut, _Homotopy 4-spheres associated to an infinite order loose cork_ ([arXiv:1901.08299](https://arxiv.org/abs/1901.08299)) * Akio Kawauchi, _Smooth homotopy 4-sphere_ ([arXiv:1911.11904](https://arxiv.org/abs/1911.11904)) * David T. Gay, _Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins_ ([arXiv:2102.12890](https://arxiv.org/abs/2102.12890)) ### Branched covers All [[PL manifold|PL]] [[4-manifolds]] are _simple_ [[branched covers]] of the [[4-sphere]]: * {#Piergallini95} [[Riccardo Piergallini]], _Four-manifolds as 4-fold branched covers of $S^4$_, Topology Volume 34, Issue 3, July 1995 (<a href="https://doi.org/10.1016/0040-9383(94)00034-I">doi:10.1016/0040-9383(94)00034-I</a>, [pdf](https://core.ac.uk/download/pdf/82379618.pdf)) * {#IoriPiergallini02} 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](https://arxiv.org/abs/math/0203087)) {#ConnesOnCohomotopy} Speculative remarks on the possible role of maps from [[spacetime]] to the [[4-sphere]] in some kind of [[quantum gravity]] via [spectral geometry](spectral+triple) (related to the [[Connes-Lott-Chamseddine-Barrett model]]) are in * {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Quanta of Geometry: Noncommutative Aspects_, Phys. Rev. Lett. 114 (2015) 9, 091302 ([arXiv:1409.2471](https://arxiv.org/abs/1409.2471)) * {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Geometry and the Quantum: Basics_, JHEP 12 (2014) 098 ([arXiv:1411.0977](https://arxiv.org/abs/1411.0977)) * {#Connes17} [[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](https://arxiv.org/abs/1703.02470), [doi:10.1007/978-3-319-64813-2](https://www.springer.com/gp/book/9783319648125)) * [[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](https://www.youtube.com/watch?v=qVqqftQ92kA)) [[!redirects 4-spheres]]</textarea> </div>  </body> </html>