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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{4-sphere} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms} [[!include manifolds and cobordisms - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{CosetSpaceStructure}{Coset space structure}\dotfill \pageref*{CosetSpaceStructure} \linebreak \noindent\hyperlink{homotopy_groups}{Homotopy groups}\dotfill \pageref*{homotopy_groups} \linebreak \noindent\hyperlink{as_part_of_the_quaternionic_hopf_fibration}{As part of the quaternionic Hopf fibration}\dotfill \pageref*{as_part_of_the_quaternionic_hopf_fibration} \linebreak \noindent\hyperlink{as_a_quotient_of_the_complex_projective_plane}{As a quotient of the complex projective plane}\dotfill \pageref*{as_a_quotient_of_the_complex_projective_plane} \linebreak \noindent\hyperlink{exotic_smooth_structures}{Exotic smooth structures}\dotfill \pageref*{exotic_smooth_structures} \linebreak \noindent\hyperlink{QuaternionAction}{$SU(2)$ action}\dotfill \pageref*{QuaternionAction} \linebreak \noindent\hyperlink{CircleAction}{Circle action}\dotfill \pageref*{CircleAction} \linebreak \noindent\hyperlink{m5brane_orbifolds}{M5-brane orbifolds}\dotfill \pageref*{m5brane_orbifolds} \linebreak \noindent\hyperlink{FreeLoopSpace}{Free and cyclic loop space}\dotfill \pageref*{FreeLoopSpace} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{branched_covers}{Branched covers}\dotfill \pageref*{branched_covers} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The [[sphere]] of [[dimension]] 4. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{CosetSpaceStructure}{}\subsubsection*{{Coset space structure}}\label{CosetSpaceStructure} As any [[sphere]], the [[4-sphere]] has the [[coset space]] [[structure]] \begin{displaymath} S^4 \simeq O(5)/O(4) \simeq SO(5)/SO(4) \simeq Spin(5)/Spin(4)\simeq Pin(5)/Pin(4). \end{displaymath} There is also this: \begin{example} \label{Sp2Sp1BySp1Sp1Sp1IsS4}\hypertarget{Sp2Sp1BySp1Sp1Sp1IsS4}{} The [[coset space]] of [[Sp(2).Sp(1)]] (\href{SpnSp1#SpnSp1}{this Def.}) by [[Sp(1)Sp(1)Sp(1)]] (\href{SpnSp1#Spin4Spin3}{this Def.}) is the [[4-sphere]]: \begin{displaymath} \frac{ Sp(2)\cdot Sp(1) } { Sp(1)Sp(1)Sp(1) } \;\simeq\; S^4 \,. \end{displaymath} This follows essentially from the [[quaternionic Hopf fibration]] and its $Sp(2)$-[[equivariant function|equivariance]]\ldots{} \end{example} (e.g. \hyperlink{BettiolMendes15}{Bettiol-Mendes 15, (3.1), (3.2), (3.3)}) \hypertarget{homotopy_groups}{}\subsubsection*{{Homotopy groups}}\label{homotopy_groups} The [[homotopy groups]] of the 4-sphere in low degree are \begin{tabular}{l|l|l|l|l|l|l|l|l|l|l|l|l|l} $k$&0&1&2&3&4&5&6&7&8&9&10&11&12\\ \hline $\pi_k(S^4)$&$\ast$&0&0&0&$\mathbb{Z}$&$\mathbb{Z}_2$&$\mathbb{Z}_2$&$\mathbb{Z} \times \mathbb{Z}_2$&$\mathbb{Z}_2^2$&$\mathbb{Z}_2^2$&$\mathbb{Z}_{24} \times \mathbb{Z}_3$&$\mathbb{Z}_{15}$&$\mathbb{Z}_2$\\ \end{tabular} \hypertarget{as_part_of_the_quaternionic_hopf_fibration}{}\subsubsection*{{As part of the quaternionic Hopf fibration}}\label{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 $\mathbb{H}$. \begin{displaymath} \itexarray{ S^3 &\hookrightarrow& S^7 \\ && \downarrow^\mathrlap{p} \\ && S^4 &\stackrel{}{\longrightarrow}& \mathbf{B} SU(2) } \end{displaymath} Here the idea is that $S^7$ may be construed as \begin{displaymath} \itexarray{ S^7 &\simeq S(\mathbb{H}^4) \\ & \simeq \{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\}, } \end{displaymath} 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 \emph{[[Hopf construction]]}, see there the section \emph{\href{Hopf+construction#HopfFibrations}{Hopf fibrations}}. By this parameterization $S^4$ is identified as $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$. \hypertarget{as_a_quotient_of_the_complex_projective_plane}{}\subsubsection*{{As a quotient of the complex projective plane}}\label{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]]. \begin{displaymath} \mathbb{C}P^2 / (-)^* \simeq S^4 \end{displaymath} \begin{itemize}% \item [[Vladimir Arnold]], \emph{Ramied covering $\mathbb{C}P^2 \to S^4$, hyperbolicity and projective topology}, Siberian Math. Journal 1988, V. 29, N 5, P.36-47 \item [[Vladimir Arnold]], \emph{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. \item [[William Massey]], \emph{The quotient space of the complex projective space under conjugation is a 4-sphere}, Geometriae Didactica 1973 \item [[Nicolaas Kuiper]], \emph{The quotient space of ℂP(2) by complex conjugation is the 4-sphere}, Mathematische Annalen, 1974 \item J.A.Hillman, \emph{An explicit formula for a branched covering from $\mathbb{C}P^2$ to $S^4$} (\href{https://arxiv.org/abs/1705.05038}{arXiv:1705.05038}) \end{itemize} \hypertarget{exotic_smooth_structures}{}\subsubsection*{{Exotic smooth structures}}\label{exotic_smooth_structures} It is open whether the 4-sphere admits an [[exotic smooth structure]]. See (\hyperlink{FreedmanGompfMorrisonWalker09}{Freedman-Gompf-Morrison-Walker 09}) for review. \hypertarget{QuaternionAction}{}\subsubsection*{{$SU(2)$ action}}\label{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 \begin{displaymath} S^4 \simeq S(\mathbb{R} \oplus \mathbb{H}) \end{displaymath} as in the discussion of the quaternionic Hopf fibration \hyperlink{HopfParameterization}{above}, then there is induced an [[action]] of the group [[special unitary group|SU(2)]] on the 4-sphere, by identifying \begin{displaymath} SU(2) \simeq S(\mathbb{Q}) \end{displaymath} and then acting by left multiplication. \hypertarget{CircleAction}{}\paragraph*{{Circle action}}\label{CircleAction} \begin{prop} \label{}\hypertarget{}{} 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: \begin{enumerate}% \item either it is the [[0-sphere]] $S^0 \hookrightarrow S^4$ \item or it has the [[rational homotopy theory|rational homotopy type]] of an even-dimensional sphere. \end{enumerate} \end{prop} (\hyperlink{FelixOpreaTanre08}{Félix-Oprea-Tanré 08, Example 7.39}) For more see at \emph{[[group actions on spheres]]}. As a special case of the $SU(2)$-action from \hyperlink{QuaternionAction}{above}, we discuss the induced circle action via the embedding \begin{displaymath} S^1 \simeq U(1) \hookrightarrow SU(2) \,. \end{displaymath} Consider the following [[circle group|circle]] [[group action on an n-sphere|group action on the 4-sphere]]: \begin{defn} \label{CircleActionOn4Sphere}\hypertarget{CircleActionOn4Sphere}{} \textbf{($SU(2)$-action on 4-sphere)} Regard \begin{displaymath} S^4 \simeq S(\mathbb{R} \oplus \mathbb{H}) \end{displaymath} 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 \begin{displaymath} SU(2) \simeq S(\mathbb{H},\cdot) \end{displaymath} In particular this restricts to an [[action]] of the [[circle group]] \begin{displaymath} S^1 \simeq U(1) \hookrightarrow SU(2) \end{displaymath} (as the [[diagonal matrices]] inside $SU(2)$) on the 4-sphere. \end{defn} 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 \begin{displaymath} S^0 \;=\; \{(\pm 1, 0,0,0,0)\} \;\in\; \mathbb{R} \oplus \mathbb{H} \end{displaymath} 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]] \begin{displaymath} S^4//S^1 \longrightarrow S^4 / S^1 \simeq S^3 \,. \end{displaymath} 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 \hyperlink{RoigSaralegiAranguren00}{Roig \& Saralegi-Aranguren 00, second page}: \begin{prop} \label{FourSphereOverThreeSphereMinimalDgModel}\hypertarget{FourSphereOverThreeSphereMinimalDgModel}{} \textbf{(\hyperlink{RoigSaralegiAranguren00}{Roig \& Saralegi-Aranguren 00, p. 2})} Write \begin{displaymath} CE(\mathfrak{l}(S^3))) = Sym^\bullet \langle \underset{\text{deg 3}}{\underbrace{h_3}} \rangle \end{displaymath} for the [[minimal Sullivan model]] of the [[3-sphere]]. Then [[rational homotopy theory|rational]] [[minimal dg-modules]] for the maps (via Def. \ref{CircleActionOn4Sphere}) \begin{displaymath} \itexarray{ S^4 \\ \downarrow \\ S^3 } \,,\phantom{AA} \itexarray{ S^4//S^1 \\ \downarrow \\ S^3 } \,,\phantom{AA} \itexarray{ S^0 \\ \downarrow \\ S^3 } \end{displaymath} as [[dg-modules]] over $CE(\mathfrak{l}(S^3))$ are given as follows, respectively: \begin{equation} \itexarray{ \text{fibration} & \itexarray{\text{vector space underlying} \\ \text{minimal dg-model}} & \itexarray{ \text{differential on} \\ \text{minimal dg-model} } \\ \itexarray{ 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. \\ \itexarray{ 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. \\ \itexarray{ 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. } \label{FourSphereAndRelatedOverThreeSphereMinimalDGModels}\end{equation} \end{prop} 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 \hyperlink{RoigSaralegiAranguren00}{Roig \& Saralegi-Aranguren 00, second page}, to bring out the pattern: \begin{tabular}{l|l} $\phantom{A}$Roig$\phantom{A}$&$\phantom{A}$here$\phantom{A}$\\ \hline $\phantom{A}a\phantom{A}$&$\phantom{A}h_3\phantom{A}$\\ $\phantom{A}1\phantom{A}$&$\phantom{A}\tilde\omega_0\phantom{A}$\\ $\phantom{A}c_{2n}\phantom{A}$&$\phantom{A}\tilde\omega_{2n+2}\phantom{A}$\\ $\phantom{A}c_{2n+1}\phantom{A}$&$\phantom{A}\omega_{2n+4}\phantom{A}$\\ $\phantom{A}e\phantom{A}$&$\phantom{A}\omega_2\phantom{A}$\\ $\phantom{A}\gamma_{2n}\phantom{A}$&$\phantom{A}\tilde\omega_{2n}\phantom{A}$\\ $\phantom{A}\gamma_{2n+1}\phantom{A}$&$\phantom{A}\omega_{2n}\phantom{A}$\\ \end{tabular} \hypertarget{m5brane_orbifolds}{}\paragraph*{{M5-brane orbifolds}}\label{m5brane_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]] \begin{displaymath} AdS_7 \times X_4 \end{displaymath} (the [[near horizon geometry]] of a [[black brane|black]] [[M5-brane]]) are all of the form \begin{displaymath} X_4 \simeq S^4//G \end{displaymath} 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 \hyperlink{QuaternionAction}{above}, and where the double slash denotes the [[homotopy quotient]] ([[orbifold quotient]]). See (\hyperlink{AFHS98}{AFHS 98, section 5.2}, \hyperlink{MF12}{MF 12, section 8.3}). \hypertarget{FreeLoopSpace}{}\subsubsection*{{Free and cyclic loop space}}\label{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 \emph{[[Sullivan models of free loop spaces]]}: \begin{example} \label{}\hypertarget{}{} Let $X = S^4$ be the [[4-sphere]]. The corresponding [[rational n-sphere]] has minimal Sullivan model \begin{displaymath} (\wedge^\bullet \langle g_4, g_7 \rangle, d) \end{displaymath} with \begin{displaymath} d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4 \,. \end{displaymath} Hence \href{Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace}{this prop.} gives for the rationalization of $\mathcal{L}S^4$ the model \begin{displaymath} ( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4} ) \end{displaymath} with \begin{displaymath} \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} \end{displaymath} and \href{Sullivan+model+of+free+loop+space#ModelForS1quotient}{this prop} gives for the rationalization of $\mathcal{L}S^4 / / S^1$ the model \begin{displaymath} ( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4 / / S^1} ) \end{displaymath} with \begin{displaymath} \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} \,. \end{displaymath} \end{example} \begin{prop} \label{}\hypertarget{}{} 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]] \begin{displaymath} \hat \mathfrak{g} \longrightarrow \mathfrak{g} \overset{\omega_2}{\longrightarrow} b \mathbb{R} \end{displaymath} which is dually modeled by \begin{displaymath} 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) \,. \end{displaymath} 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 $(A_X,d_X)$: \begin{displaymath} CE(\mathfrak{l}X) = (A_X,d_X) \,. \end{displaymath} Then there is an [[isomorphism]] of [[hom-sets]] \begin{displaymath} 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 ) ) \,, \end{displaymath} with $\mathfrak{l}(S^4)$ from \href{Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace}{this prop.} and $\mathfrak{l}(\mathcal{L}S^4 //S^1)$ from \href{Sullivan+model+of+free+loop+space#ModelForS1quotient}{this prop.}, where on the right we have homs in the [[slice category|slice]] over the [[line Lie n-algebra|line Lie 2-algebra]], via \href{Sullivan+model+of+free+loop+space#ModelForS1quotient}{this prop.} Moreover, this isomorphism takes \begin{displaymath} \hat \mathfrak{g} \overset{(g_4, g_7)}{\longrightarrow} \mathfrak{l}(S^4) \end{displaymath} to \begin{displaymath} \itexarray{ \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} } \,, \end{displaymath} where \begin{displaymath} \omega_4 = g_4 - h_3 \wedge e \;\,, \;\;\; h_7 = g_7 + \omega_6 \wedge e \end{displaymath} 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. \end{prop} This is observed in (\href{Sullivan+model+of+free+loop+space#FiorenzaSatiSchreiber16}{FSS 16}, \hyperlink{FSS16b}{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 [[schreiber:The brane bouquet]]). \begin{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 \begin{displaymath} d g_4 = 0 \,,\;\;\; d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4 \end{displaymath} 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{displaymath} \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} \end{displaymath} as well as \begin{displaymath} \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} \end{displaymath} \end{proof} The [[unit of an adjunction|unit]] of the [[double dimensional reduction]]-[[adjunction]] \begin{displaymath} \infty Grpd \underoverset {\underset{\mathcal{L}(-)/S^1}{\longrightarrow}} {\overset{hofib}{\longleftarrow}} {\bot} \infty Grpd_{/B S^1} \end{displaymath} (\href{double+dimensional+reduction#GeneralReduction}{this prop.}) applied to the $S^1$-[[principal infinity-bundle]] \begin{displaymath} \itexarray{ S^4 \\ {}^{\mathllap{hofib(c)}}\downarrow \\ S^4 / S^1 &\underset{c}{\longrightarrow}& B S^1 } \end{displaymath} is a natural map \begin{displaymath} S^4/S^1 \longrightarrow \mathcal{L}(S^4)/S^1 \end{displaymath} from the [[homotopy quotient]] by the [[circle action]] (def. \ref{CircleActionOn4Sphere}), to the [[cyclic loop space]] of the 4-sphere. \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[2-sphere]] \item [[3-sphere]] \item [[6-sphere]] \item [[7-sphere]] \item [[n-sphere]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item [[Michael Freedman]], [[Robert Gompf]], [[Scott Morrison]], [[Kevin Walker]], \emph{Man and machine thinking about the smooth 4-dimensional Poincar\'e{} conjecture}, Quantum Topology, Volume 1, Issue 2 (2010), pp. 171-208 (\href{http://arxiv.org/abs/0906.5177}{arXiv:0906.5177}) \item [[Agustí Roig]], [[Martintxo Saralegi-Aranguren]], \emph{Minimal Models for Non-Free Circle Actions}, Illinois Journal of Mathematics, volume 44, number 4 (2000) (\href{https://arxiv.org/abs/math/0004141}{arXiv:math/0004141}) \item [[Bobby Acharya]], [[José Figueroa-O'Farrill]], [[Chris Hull]], B. Spence, \emph{Branes at conical singularities and holography} , Adv. Theor. Math. Phys. 2 (1998) 1249--1286 \item [[Yves Félix]], John Oprea, [[Daniel Tanré]], \emph{Algebraic Models in Geometry}, Oxford University Press 2008 \item Paul de Medeiros, [[José Figueroa-O'Farrill]], \emph{Half-BPS M2-brane orbifolds}, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (\href{http://arxiv.org/abs/1007.4761}{arXiv:1007.4761}, \href{https://projecteuclid.org/euclid.atmp/1408561553}{Euclid}) \item Renato G. Bettiol, Ricardo A. E. Mendes, \emph{Flag manifolds with strongly positive curvature}, Math. Z. 280 (2015), no. 3-4, 1031-1046 (\href{https://arxiv.org/abs/1412.0039}{arXiv:1412.0039}) \item Selman Akbulut, \emph{Homotopy 4-spheres associated to an infinite order loose cork} (\href{https://arxiv.org/abs/1901.08299}{arXiv:1901.08299}) \end{itemize} \hypertarget{branched_covers}{}\subsubsection*{{Branched covers}}\label{branched_covers} All [[PL manifold|PL]] [[4-manifolds]] are \emph{simple} [[branched covers]] of the [[4-sphere]]: \begin{itemize}% \item [[Riccardo Piergallini]], \emph{Four-manifolds as 4-fold branched covers of $S^4$}, Topology Volume 34, Issue 3, July 1995 (, \href{https://core.ac.uk/download/pdf/82379618.pdf}{pdf}) \item Massimiliano Iori, [[Riccardo Piergallini]], \emph{4-manifolds as covers of the 4-sphere branched over non-singular surfaces}, Geom. Topol. 6 (2002) 393-401 (\href{https://arxiv.org/abs/math/0203087}{arXiv:math/0203087}) \end{itemize} Speculative remarks on the possible role of maps from [[spacetime]] to the [[4-sphere]] in some kind of [[quantum gravity]] via \href{spectral+triple}{spectral geometry} (related to the [[Connes-Lott-Chamseddine-Barrett model]]) are in \begin{itemize}% \item [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, \emph{Quanta of Geometry: Noncommutative Aspects}, Phys. Rev. Lett. 114 (2015) 9, 091302 (\href{https://arxiv.org/abs/1409.2471}{arXiv:1409.2471}) \item [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, \emph{Geometry and the Quantum: Basics}, JHEP 12 (2014) 098 (\href{https://arxiv.org/abs/1411.0977}{arXiv:1411.0977}) \item [[Alain Connes]], section 4 of \emph{Geometry and the Quantum}, in \emph{Foundations of Mathematics and Physics One Century After Hilbert}, Springer 2018. 159-196 (\href{https://arxiv.org/abs/1703.02470}{arXiv:1703.02470}, \href{https://www.springer.com/gp/book/9783319648125}{doi:10.1007/978-3-319-64813-2}) \item [[Alain Connes]], from 58:00 to 1:25:00 in \emph{Why Four Dimensions and the Standard Model Coupled to Gravity - A Tentative Explanation From the New Geometric Paradigm of NCG}, talk at IHES, 2017 (\href{https://www.youtube.com/watch?v=qVqqftQ92kA}{video recording}) \end{itemize} [[!redirects 4-spheres]] \end{document}