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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*{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{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{finitedimensional_spheres}{Finite-dimensional spheres}\dotfill \pageref*{finitedimensional_spheres} \linebreak \noindent\hyperlink{infinite_dimensional_spheres}{Infinite dimensional spheres}\dotfill \pageref*{infinite_dimensional_spheres} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{basic}{Basic}\dotfill \pageref*{basic} \linebreak \noindent\hyperlink{LabelCosetSpaceStructure}{Coset space structure}\dotfill \pageref*{LabelCosetSpaceStructure} \linebreak \noindent\hyperlink{parallelizability}{Parallelizability}\dotfill \pageref*{parallelizability} \linebreak \noindent\hyperlink{branched_covers}{Branched covers}\dotfill \pageref*{branched_covers} \linebreak \noindent\hyperlink{iterated_loop_spaces}{Iterated loop spaces}\dotfill \pageref*{iterated_loop_spaces} \linebreak \noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{formalization}{Formalization}\dotfill \pageref*{formalization} \linebreak \noindent\hyperlink{group_actions_on_spheres}{Group actions on spheres}\dotfill \pageref*{group_actions_on_spheres} \linebreak \noindent\hyperlink{geometric_structures_on_spheres}{Geometric structures on spheres}\dotfill \pageref*{geometric_structures_on_spheres} \linebreak \noindent\hyperlink{embeddings_of_spheres}{Embeddings of spheres}\dotfill \pageref*{embeddings_of_spheres} \linebreak \noindent\hyperlink{iterated_loop_spaces_2}{Iterated loop spaces}\dotfill \pageref*{iterated_loop_spaces_2} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{finitedimensional_spheres}{}\subsubsection*{{Finite-dimensional spheres}}\label{finitedimensional_spheres} \begin{defn} \label{}\hypertarget{}{} The $n$-dimensional unit \textbf{sphere} , or simply \textbf{$n$-sphere}, is the [[topological space]] given by the [[subset]] of the $(n+1)$-dimensional [[Cartesian space]] $\mathbb{R}^{n+1}$ consisting of all points $x$ whose distance from the origin is $1$ \begin{displaymath} S^n = \{ x: \mathbb{R}^{n+1} \;|\; \|x\| = 1 \} \,. \end{displaymath} The $n$-dimensional sphere of radius $r$ is \begin{displaymath} S^n_r = \{ x: \mathbb{R}^{n+1} \;|\; \|x\| = r \} . \end{displaymath} Topologically, this is equivalent ([[homeomorphism|homeomorphic]]) to the unit sphere for $r \gt 0$, or a [[point]] for $r = 0$. This is naturally also a [[smooth manifold]] of [[dimension]] $n$, with the [[smooth structure]] induced from the standard sooth structure on $\mathbb{R}${\tt \symbol{94}}n. \end{defn} \hypertarget{infinite_dimensional_spheres}{}\subsubsection*{{Infinite dimensional spheres}}\label{infinite_dimensional_spheres} One can also talk about a sphere in an arbitrary (possibly infinite-dimensional) [[normed vector space]] $V$: \begin{displaymath} S(V) = \{ x: V \;\text{such that}\; {\|x\|} = 1 \} . \end{displaymath} If a [[LCTVS|locally convex topological vector space]] admits a continuous linear injection into a [[normed vector space]], this can be used to define its sphere. If not, one can still define the sphere as a \emph{quotient} of the space of non-zero vectors under the scalar action of $(0,\infty)$. Homotopy theorists define $S^\infty$ to be the sphere in the (incomplete) [[normed vector space]] (traditionally with the $l^2$ norm) of infinite [[sequence]]s almost all of whose values are $0$, which is the [[directed colimit]] of the $S^n$: \begin{displaymath} S^{-1} \hookrightarrow S^0 \hookrightarrow S^1 \hookrightarrow S^2 \hookrightarrow \cdots S^\infty . \end{displaymath} In themselves, these provide nothing new to [[homotopy theory]], as they are at least weakly contractible and usually [[contractible space|contractible]]. However, they are a very useful source of big contractible spaces and so are often used as a starting point for making concrete models of [[classifying spaces]]. If the vector space is a [[shift space]], then contractibility is straightforward to prove. \begin{theorem} \label{theorem}\hypertarget{theorem}{} Let $V$ be a [[shift space]] of some order. Let $S V$ be its sphere (either via a norm or as the quotient of non-zero vectors). Then $S V$ is contractible. \end{theorem} \begin{proof} Let $T \colon V \to V$ be a [[shift map]]. The idea is to homotop the sphere onto the image of $T$, and then down to a point. It is simplest to start with the non-zero vectors, $V \setminus \{0\}$. As $T$ is injective, it restricts to a map from this space to itself which commutes with the scalar action of $(0,\infty)$. Define a homotopy $H \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $H_t(v) = (1 - t)v + t T v$. It is clear that, assuming it is well-defined, it is a homotopy from the identity to $T$. To see that it is well-defined, we need to show that $H_t(v)$ is never zero. The only place where it could be zero would be on an eigenvector of $T$, but as $T$ is a [[shift map]] then it has none. As $T$ is a [[shift map]], it is not surjective and so we can pick some $v_0$ not in its image. Then we define a homotopy $G \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $G_t(v) = (1 - t)T v + t v_0$. As $v_0$ is not in the image of $T$, this is well-defined on $V \setminus \{0\}$. Combining these two homotopies results in the desired contraction of $V \setminus \{0\}$. If $V$ admits a suitable function defining a spherical subset (such as a norm) then we can modify the above to a contraction of the spherical subset simply by dividing out by this function. If not, as the homotopies above all commute with the scalar action of $(0,\infty)$, they descend to the definition of the sphere as the quotient of $V \setminus \{0\}$. \end{proof} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{basic}{}\subsubsection*{{Basic}}\label{basic} \begin{itemize}% \item The $n$-sphere is the [[boundary]] of the $(n+1)$-[[ball]]. \item These spheres, or rather their underlying [[topological spaces]] or [[simplicial set]]s, are fundamental in (ungeneralised) [[homotopy theory]]. In a sense, [[Whitehead's theorem]] says that these are all that you need; no further [[generalized (Eilenberg-Steenrod) homotopy theory|generalised homotopy theory]] (in a sense [[Eckmann–Hilton duality|dual]] to [[Eilenberg–Steenrod cohomology theory]]) is needed. \item [[positive dimension spheres are H-cogroup objects]], and this is the origin of the [[group]] structure on [[homotopy groups]]). \end{itemize} \hypertarget{LabelCosetSpaceStructure}{}\subsubsection*{{Coset space structure}}\label{LabelCosetSpaceStructure} \begin{prop} \label{nSphereAsCosetSpace}\hypertarget{nSphereAsCosetSpace}{} For $n \in \mathbb{N}$ the [[n-spheres]] are [[coset spaces]] of [[orthogonal groups]] \begin{displaymath} S^n \;\simeq\; O(n+1)/O(n) \,. \end{displaymath} Similarly for the corresponding [[special orthogonal groups]] \begin{displaymath} S^n \;\simeq\; SO(n+1)/SO(n) \end{displaymath} and [[spin groups]] \begin{displaymath} S^n \;\simeq\; Spin(n+1)/Spin(n) \end{displaymath} and [[pin groups]] \begin{displaymath} S^n \;\simeq\; Pin(n+1)/Pin(n) \,. \end{displaymath} \end{prop} \begin{proof} Fix a [[unit vector]] in $\mathbb{R}^{n+1}$. Then its [[orbit]] under the defining $O(n+1)$-[[action]] on $\mathbb{R}^{n+1}$ is clearly the canonical embedding $S^n \hookrightarrow \mathbb{R}^{n+1}$. But precisely the subgroup of $O(n+1)$ that consists of rotations around the axis formed by that unit vector [[stabilizer group|stabilizes]] it, and that subgroup is isomorphic to $O(n)$, hence $S^n \simeq O(n+1)/O(n)$. \end{proof} Similarly, the analogous argument for [[unit spheres]] inside (the [[real vector spaces]] underlying) [[complex vector spaces]], we have \begin{prop} \label{OddDimSphereAsSpecialUnitaryCoset}\hypertarget{OddDimSphereAsSpecialUnitaryCoset}{} For $k \in \mathbb{N}$ the [[n-sphere|(2k+1)-sphere]] $S^{2k+1}$ is the [[coset space]] of [[special unitary groups]]: \begin{displaymath} S^{2k+1} \;\simeq\; SU(k+1)/SU(k) \,. \end{displaymath} \end{prop} And still similarly, the analogous argument for [[unit spheres]] inside (the [[real vector spaces]] underlying) [[quaternionic vector spaces]], we have \begin{prop} \label{SphereAsSymplecticUnitaryCoset}\hypertarget{SphereAsSymplecticUnitaryCoset}{} For $k \in \mathbb{N}$, $k \geq 1$ the [[n-sphere|(4k-1)-sphere]] $S^{4k-1}$ is the [[coset space]] of [[quaternionic unitary groups]]: \begin{displaymath} S^{4k-1} \;\simeq\; Sp(k)/Sp(k-1) \,. \end{displaymath} \end{prop} Generally: \begin{prop} \label{TransitiveEffectiveActionsOfConnectedLieGroupsOnSpheres}\hypertarget{TransitiveEffectiveActionsOfConnectedLieGroupsOnSpheres}{} The [[connected topological space|connected]] [[Lie groups]] with [[effective action|effective]] [[transitive actions]] on [[n-spheres]] are precisely (up to [[isomorphism]]) the following: \begin{itemize}% \item [[SO(n)]] \item [[U(n)]] \item [[SU(n)]] \item [[Sp(n)]] \item [[Sp(n).Sp(1)|Sp(n).SO(2)]] \item [[Sp(n).Sp(1)]] \item [[G2]] \item [[Spin(7)]] \item [[Spin(9)]] \end{itemize} with [[coset spaces]] \begin{displaymath} \begin{aligned} SO(n)/SO(n-1) & \simeq S^{n-1} \\ U(n)/U(n-1) & \simeq S^{2n-1} \\ SU(n)/SU(n-1) & \simeq S^{2n-1} \\ Sp(n)/Sp(n-1) & \simeq S^{4n-1} \\ Sp(n)\cdot SO(2)/Sp(n-1)\cdot SO(2) & \simeq S^{4n-1} \\ Sp(n)\cdot Sp(1)/Sp(n-1)\cdot Sp(1) & \simeq S^{4n-1} \\ G_2/SU(3) & \simeq S^6 \\ Spin(7)/G_2 & \simeq S^7 \\ Spin(9)/Spin(7) & \simeq S^{15} \end{aligned} \end{displaymath} \end{prop} This goes back to \hyperlink{MontogomerySamelson43}{Montogomery-Samelson 43}, see\newline \hyperlink{GrayGreen70}{Gray-Green 70, p. 1-2} (also e.g \hyperlink{BorelSerre53}{Borel-Serre 53, 17.1}) \begin{remark} \label{}\hypertarget{}{} The [[isomorphisms]] in Prop. \ref{nSphereAsCosetSpace} and Prop. \ref{OddDimSphereAsSpecialUnitaryCoset} above hold in the [[category]] of [[topological spaces]] ([[homeomorphisms]]), but in fact also in the [[category]] of [[smooth manifolds]] ([[diffeomorphisms]]) and even in the [[category]] of [[Riemannian manifolds]] ([[isometries]]). But the other [[coset space]] realizations of some [[n-spheres]] in Prop. \ref{TransitiveEffectiveActionsOfConnectedLieGroupsOnSpheres} are [[homeomorphisms]], but not necessarily [[isometries]] (``[[squashed spheres]]''). There is also a [[double coset space]] realization which is not even a [[diffeomorphisms]] (``[[exotic sphere]]'', the [[Gromoll-Meyer sphere]]). For more see \emph{\hyperlink{7-sphere#CosetSpaceRealization}{7-sphere -- Coset space realization}}. \end{remark} [[!include coset space structure on n-spheres -- table]] $\backslash$linebreak \hypertarget{parallelizability}{}\subsubsection*{{Parallelizability}}\label{parallelizability} \begin{itemize}% \item Precisely four spheres are [[parallelizable]], and three of these are so via [[Lie group]] structure (hence are the only spheres with Lie group structure) (see at \emph{[[Hopf invariant one theorem]]}): \begin{itemize}% \item $S^0$ (the [[group of order two]], the group of units of the [[real numbers]]); \item $S^1$ (the [[circle group]], the group of unit [[complex numbers]]); \item $S^3$ (the [[special unitary group]] $SU(2)$, the group of unit [[quaternions]]); \item $S^7$ (the [[Moufang loop]] of unit [[octonions]]) \end{itemize} \end{itemize} \hypertarget{branched_covers}{}\subsubsection*{{Branched covers}}\label{branched_covers} Every $n$-[[dimension|dimensional]] [[PL manifold]] admits a [[branched covering]] of the [[n-sphere]] (\href{branched+cover#Alexander20}{Alexander 20}). By the [[Riemann existence theorem]], every [[connected topological space|connected]] [[compact topological space|compact]] [[Riemann surface]] admits the [[structure]] of a branched cover by a [[holomorphic function]] to the [[Riemann sphere]]. See \href{branched+cover+of+the+Riemann+sphere#RiemannSurfaces}{there} at \emph{[[branched cover of the Riemann sphere]]}. $\backslash$begin\{center\} $\backslash$end\{center\} \begin{quote}% graphics grabbed from \href{branched+cover#ChamseddineConnesMukhanov14}{Chamseddine-Connes-Mukhanov 14, Figure 1}, \href{branched+cover#Connes17}{Connes 17, Figure 11} \end{quote} For [[3-manifolds]] branched covering the [[3-sphere]] see (\href{branched+cover#Montesinos74}{Montesinos 74}). All [[PL manifold|PL]] [[4-manifolds]] are \emph{simple} branched covers of the [[4-sphere]] (\hyperlink{Piergallini95}{Piergallini 95}, \href{branched+cover#IoriPiergallini02}{Iori-Piergallini 02}). But the [[n-torus]] for $n \geq 3$ is \emph{not a [[cyclic group|cyclic]]} branched over of the [[n-sphere]] (\href{branched+cover#HirschNeumann75}{Hirsch-Neumann 75}) \hypertarget{iterated_loop_spaces}{}\subsubsection*{{Iterated loop spaces}}\label{iterated_loop_spaces} \begin{prop} \label{RationalCohomologyOfIteratedLoopSpaceOf2kSphere}\hypertarget{RationalCohomologyOfIteratedLoopSpaceOf2kSphere}{} \textbf{([[rational cohomology]] of [[iterated loop space]] of the [[n-sphere|2k-sphere]])} Let \begin{displaymath} 1 \leq D \lt n = 2k \in \mathbb{N} \end{displaymath} (hence two [[positive number|positive]] [[natural numbers]], one of them required to be [[even number|even]] and the other required to be smaller than the first) and consider the [[iterated loop space|D-fold loop space]] $\Omega^D S^n$ of the [[n-sphere]]. Its [[rational cohomology|rational]] [[cohomology ring]] is the [[free construction|free]] [[graded-commutative algebra]] over $\mathbb{Q}$ on one [[generators and relations|generator]] $e_{n-D}$ of degree $n - D$ and one generator $a_{2n - D - 1}$ of degree $2n - D - 1$: \begin{displaymath} H^\bullet \big( \Omega^D S^n , \mathbb{Q} \big) \;\simeq\; \mathbb{Q}\big[ e_{n - D}, a_{2n - D - 1} \big] \,. \end{displaymath} \end{prop} (\hyperlink{KallelSjerve99}{Kallel-Sjerve 99, Prop. 4.10}) \hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples} \begin{itemize}% \item The $(-1)$-sphere is the [[empty space]]. \item The [[0-sphere]] is the [[disjoint union]] of two [[points]]. \item The [[1-sphere]] is the [[circle]]. \item The [[2-sphere]] is usual sphere from ordinary geometry. This canonically carries the structure of a [[complex manifold]] which makes it the [[Riemann sphere]]. \item The [[3-sphere]] and [[4-sphere]] and [[6-sphere]] and [[7-sphere]] are interesting, too. \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[round sphere]] \item [[hemisphere]], [[equator]] \item [[unit sphere]] \item [[stereographic projection]] \item [[sphere fiber bundle]] \item [[polar coordinates]] \item [[Reeb sphere theorem]] \item [[homotopy groups of spheres]] \item [[rational n-sphere]] \item [[sphere spectrum]] \item [[spherical fibration]] \item [[geometric quantization of the 2-sphere]] \item [[representation sphere]] \item [[motivic sphere]] \item [[group actions on spheres]] \item [[homology sphere]] \item [[homotopy sphere]] \item [[sphere packing]] \item The [[non-abelian cohomology|non-abelian]] [[generalized cohomology theory]] [[representable functor|represented]] by [[n-spheres]] is [[Cohomotopy cohomology theory]]. \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{formalization}{}\subsubsection*{{Formalization}}\label{formalization} Axiomatization of the [[homotopy type]] of the 1-sphere (the [[circle]]) and the 2-sphere, as [[higher inductive types]], is in \begin{itemize}% \item [[Univalent Foundations Project]], section 6.4 of \emph{[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]} \end{itemize} Visualization of the idea of the construction for the 2-sphere is in \begin{itemize}% \item [[Andrej Bauer]], \emph{HoTT $S^2$} (\href{https://vimeo.com/119606901}{video}) \end{itemize} \hypertarget{group_actions_on_spheres}{}\subsubsection*{{Group actions on spheres}}\label{group_actions_on_spheres} Discussion of [[free actions|free]] [[group actions on spheres]] by [[finite groups]] includes \begin{itemize}% \item [[C. T. C. Wall]], \emph{Free actions of finite groups on spheres}, Proceedings of Symposia in Pure Mathematics, Volume 32, 1978 (\href{http://www.maths.ed.ac.uk/~aar/papers/wall7.pdf}{pdf}) \item [[Alejandro Adem]], \emph{Constructing and deconstructing group actions} (\href{http://arxiv.org/abs/math/0212280}{arXiv:0212280}) \end{itemize} The subgroups of [[special orthogonal group|SO(8)]] which act freely on $S^7$ have been classified in \begin{itemize}% \item J. A. Wolf, \emph{Spaces of constant curvature}, Publish or Perish, Boston, Third ed., 1974 \end{itemize} and lifted to actions of [[Spin group|Spin(8)]] in \begin{itemize}% \item S. Gadhia, \emph{Supersymmetric quotients of M-theory and supergravity backgrounds}, PhD thesis, School of Mathematics, University of Edinburgh, 2007 \end{itemize} Discussion of [[transitive actions]] on $n$-spheres by [[compact Lie groups]]: \begin{itemize}% \item Deane Montgomery, Hans Samelson, \emph{Transformation Groups of Spheres}, Annals of Mathematics Second Series, Vol. 44, No. 3 (Jul., 1943), pp. 454-470 (\href{https://www.jstor.org/stable/1968975}{jstor:1968975}) \item [[Alfred Gray]], Paul S. Green, \emph{Sphere transitive structures and the triality automorphism}, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (\href{https://projecteuclid.org/euclid.pjm/1102976640}{euclid:1102976640}) \end{itemize} Further discussion of these actions is in \begin{itemize}% \item Paul de Medeiros, [[José Figueroa-O'Farrill]], Sunil Gadhia, [[Elena Méndez-Escobar]], \emph{Half-BPS quotients in M-theory: ADE with a twist}, JHEP 0910:038,2009 (\href{http://arxiv.org/abs/0909.0163}{arXiv:0909.0163}, \href{http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf}{pdf slides}) \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}) \end{itemize} where they are related to the [[black brane|black]] [[M2-brane]] [[BPS state|BPS]]-solutions of [[11-dimensional supergravity]] at [[ADE-singularities]]. See also the [[ADE classification]] of such actions on the [[7-sphere]] (as discussed there) \hypertarget{geometric_structures_on_spheres}{}\subsubsection*{{Geometric structures on spheres}}\label{geometric_structures_on_spheres} Coset space structures on spheres: \begin{itemize}% \item [[Armand Borel]], [[Jean-Pierre Serre]], \emph{Groupes de Lie et Puissances Reduites de Steenrod}, American Journal of Mathematics, Vol. 75, No. 3 (Jul., 1953), pp. 409-448 (\href{https://www.jstor.org/stable/2372495}{jstor:2372495}) \end{itemize} The following to be handled with care: \begin{itemize}% \item [[Michael Atiyah]], \emph{The non-existent complex 6-sphere}, \href{https://arxiv.org/abs/1610.09366}{arxiv/1610.09366} \end{itemize} \hypertarget{embeddings_of_spheres}{}\subsubsection*{{Embeddings of spheres}}\label{embeddings_of_spheres} The ([[isotopy]] [[equivalence class|class]] of an) [[embedding of differentiable manifolds|embedding]] of a [[circle]] (1-sphere) into the [[3-sphere]] is a \emph{[[knot]]}. Discussion of embeddings of spheres of more general dimensions into each other: \begin{itemize}% \item [[André Haefliger]], \emph{Differentiable Embeddings of $S^n$ in $S^{n+q}$ for $q \gt 2$}, Annals of Mathematics Second Series, Vol. 83, No. 3 (May, 1966), pp. 402-436 (\href{https://www.jstor.org/stable/1970475}{jstor:1970475}) \end{itemize} \hypertarget{iterated_loop_spaces_2}{}\subsubsection*{{Iterated loop spaces}}\label{iterated_loop_spaces_2} \begin{itemize}% \item [[Sadok Kallel]], [[Denis Sjerve]], \emph{On Brace Products and the Structur eof Fibrations with Section}, 1999 (\href{https://www.math.ubc.ca/~sjer/brace.pdf}{pdf}, [[KallelSjerv99.pdf:file]]) \end{itemize} [[!redirects n-sphere]] [[!redirects n-spheres]] [[!redirects spheres]] [[!redirects infinite sphere]] [[!redirects 1-sphere]] [[!redirects 1-spheres]] [[!redirects 2-sphere]] [[!redirects 2-spheres]] \end{document}