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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*{coset} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory} [[!include group theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{internal_to_a_general_category}{Internal to a general category}\dotfill \pageref*{internal_to_a_general_category} \linebreak \noindent\hyperlink{internal_to_}{Internal to $Set$}\dotfill \pageref*{internal_to_} \linebreak \noindent\hyperlink{for_lie_groups_and_klein_geometry}{For Lie groups and Klein geometry}\dotfill \pageref*{for_lie_groups_and_klein_geometry} \linebreak \noindent\hyperlink{for_groups}{For $\infty$-groups}\dotfill \pageref*{for_groups} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{for_normal_subgroups}{For normal subgroups}\dotfill \pageref*{for_normal_subgroups} \linebreak \noindent\hyperlink{QuotientMaps}{Quotient maps}\dotfill \pageref*{QuotientMaps} \linebreak \noindent\hyperlink{as_a_homotopy_fiber}{As a homotopy fiber}\dotfill \pageref*{as_a_homotopy_fiber} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{spheres}{$n$-Spheres}\dotfill \pageref*{spheres} \linebreak \noindent\hyperlink{QuotientMapsOfCosetSpaces}{Sequences of coset spaces}\dotfill \pageref*{QuotientMapsOfCosetSpaces} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} Given a [[group]] $G$ and a [[subgroup]] $H$, then their \emph{coset object} is the [[quotient]] $G/H$, hence the set of [[equivalence classes]] of elements of $G$ where two are regarded as equivalent if they differ by right multiplication with an element in $H$. If $G$ is a [[topological group]], then the quotient is a [[topological space]] and usually called the \emph{coset space}. This is in particular a [[homogeneous space]], see there for more. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{internal_to_a_general_category}{}\subsubsection*{{Internal to a general category}}\label{internal_to_a_general_category} In a category $C$, for $G$ a [[group object]] and $H \hookrightarrow G$ a [[subgroup|subgroup object]], the left/right \emph{object of cosets} is the [[orbit|object of orbits]] of $G$ under left/right multiplication by $H$. Explicitly, the left coset space $G/H$ [[coequalizes]] the parallel morphisms \begin{displaymath} H \times G \underoverset{\mu}{proj_G}\rightrightarrows G \end{displaymath} where $\mu$ is (the inclusion $H\times G \hookrightarrow G\times G$ composed with) the group multiplication. Simiarly, the right coset space $H\backslash G$ [[coequalizes]] the parallel morphisms \begin{displaymath} G \times H \underoverset{proj_G}{\mu}\rightrightarrows G \end{displaymath} \hypertarget{internal_to_}{}\subsubsection*{{Internal to $Set$}}\label{internal_to_} Specializing the above definition to the case where $C$ is the well-pointed topos $Set$, given an element $g$ of $G$, its orbit $g H$ is an element of $G/H$ and is called a \emph{left coset}. Using [[comprehension]], we can write \begin{displaymath} G/H = \{g H | g \in G\} \end{displaymath} Similarly there is a coset on the right $H \backslash G$. \hypertarget{for_lie_groups_and_klein_geometry}{}\subsubsection*{{For Lie groups and Klein geometry}}\label{for_lie_groups_and_klein_geometry} If $H \hookrightarrow G$ is an inclusion of [[Lie groups]] then the quotient $G/H$ is also called a \emph{[[Klein geometry]]}. \hypertarget{for_groups}{}\subsubsection*{{For $\infty$-groups}}\label{for_groups} More generally, given an [[(∞,1)-topos]] $\mathbf{H}$ and a [[homomorphism]] of [[∞-group]] ojects $H \to G$, hence equivalently a morphism of their [[deloopings]] $\mathbf{B}H \to \mathbf{B}G$, then the [[homotopy quotient]] $G/H$ is given by the [[homotopy fiber]] of this map \begin{displaymath} \itexarray{ G/H &\longrightarrow& \mathbf{B}H \\ && \downarrow \\ && \mathbf{B}G } \,. \end{displaymath} See at \emph{[[∞-action]]} for more on this definition. See at \emph{[[higher Klein geometry]]} and \emph{[[higher Cartan geometry]]} for the corresponding concepts of [[higher geometry]]. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{for_normal_subgroups}{}\subsubsection*{{For normal subgroups}}\label{for_normal_subgroups} The coset inherits the structure of a group if $H$ is a [[normal subgroup]]. Unless $G$ is abelian, considering both left and right coset spaces provide different information. \hypertarget{QuotientMaps}{}\subsubsection*{{Quotient maps}}\label{QuotientMaps} \begin{prop} \label{QuotientProjectionForCompactLieGroupActingFreelyOnManifoldIsPrincipa}\hypertarget{QuotientProjectionForCompactLieGroupActingFreelyOnManifoldIsPrincipa}{} For $X$ a [[smooth manifold]] and $G$ a [[compact Lie group]] equipped with a [[free action|free]] smooth [[action]] on $X$, then the [[quotient]] [[projection]] \begin{displaymath} X \longrightarrow X/G \end{displaymath} is a $G$-[[principal bundle]] (hence in particular a [[Serre fibration]]). \end{prop} This is originally due to (\hyperlink{Gleason50}{Gleason 50}). See e.g. (\hyperlink{Cohen}{Cohen, theorem 1.3}) \begin{cor} \label{QuotientProjectionForCompactLieSubgroupIsPrincipal}\hypertarget{QuotientProjectionForCompactLieSubgroupIsPrincipal}{} For $G$ a [[Lie group]] and $H \subset G$ a [[compact Lie group|compact]] [[subgroup]], then the [[coset]] [[quotient]] [[projection]] \begin{displaymath} G \longrightarrow G/H \end{displaymath} is an $H$-[[principal bundle]] (hence in particular a [[Serre fibration]]). \end{cor} This is originally due to (\hyperlink{Samelson41}{Samelson 41}). \begin{prop} \label{ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup}\hypertarget{ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup}{} For $G$ a [[compact Lie group]] and $K \subset H \subset G$ [[closed subspace|closed]] [[subgroups]], then the [[projection]] map \begin{displaymath} p \;\colon\; G/K \longrightarrow G/H \end{displaymath} is a locally trivial $H/K$-[[fiber bundle]] (hence in particular a [[Serre fibration]]). \end{prop} \begin{proof} Observe that the projection map in question is equivalently \begin{displaymath} G \times_H (H/K) \longrightarrow G/H \,, \end{displaymath} (where on the left we form the [[Cartesian product]] and then divide out the [[diagonal action]] by $H$). This exhibits it as the $H/K$-[[fiber bundle]] [[associated bundle|associated]] to the $H$-[[principal bundle]] of corollary \ref{QuotientProjectionForCompactLieSubgroupIsPrincipal}. \end{proof} \hypertarget{as_a_homotopy_fiber}{}\subsubsection*{{As a homotopy fiber}}\label{as_a_homotopy_fiber} \begin{remark} \label{}\hypertarget{}{} In [[geometric homotopy theory]] (in an [[(∞,1)-topos]]), for $H \longrightarrow G$ any homomorphisms of [[∞-group]] objects, then the natural projection $G \longrightarrow G/H$, generally realizes $G$ as an $H$-[[principal ∞-bundle]] over $G/H$. This is exhibited by a [[homotopy pullback]] of the form \begin{displaymath} \itexarray{ G & \longrightarrow &* \\ \downarrow && \downarrow \\ G/H &\longrightarrow& \mathbf{B}H } \,. \end{displaymath} where $\mathbf{B}H$ is the [[delooping|delooping groupoid]] of $H$. This also equivalently exhibits the [[∞-action]] of $H$ on $G$ (see there for more). By the [[pasting law]] for [[homotopy pullbacks]] then we get the [[homotopy pullback]] \begin{displaymath} \itexarray{ G/H & \longrightarrow &\mathbf{B}H \\ \downarrow && \downarrow \\ * & \longrightarrow & \mathbf{B}G } \end{displaymath} which exhibits the coset as the [[homotopy fiber]] of $\mathbf{B}H \to \mathbf{B}G$. \end{remark} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{spheres}{}\subsubsection*{{$n$-Spheres}}\label{spheres} \begin{example} \label{nSphereAsCosetSpace}\hypertarget{nSphereAsCosetSpace}{} The [[n-spheres]] are coset spaces of [[orthogonal groups]]: \begin{displaymath} S^n \simeq O(n+1)/O(n) \,. \end{displaymath} The odd-dimensional spheres are also coset spaces of [[unitary groups]]: \begin{displaymath} S^{2n+1} \simeq U(n+1)/U(n) \end{displaymath} \end{example} \begin{proof} Regarding the first statement: 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 $S^n \simeq O(n+1)/O(n)$. The second statement follows by the same kind of reasoning: Clearly $U(n+1)$ [[transitive action|acts transitively]] on the [[unit sphere]] $S^{2n+1}$ in $\mathbb{C}^{n+1}$. It remains to see that its [[stabilizer subgroup]] of any point on this sphere is $U(n)$. If we take the point with [[coordinates]] $(1,0, 0, \cdots,0)$ and regard elements of $U(n+1)$ as [[matrices]], then the stabilizer subgroup consists of matrices of the block diagonal form \begin{displaymath} \left( \itexarray{ 1 & \vec 0 \\ \vec 0 & A } \right) \end{displaymath} where $A \in U(n)$. \end{proof} There are also various exceptional realizations of spheres as coset spaces. For instance: [[!include coset space structure on n-spheres -- table]] $\backslash$linebreak \hypertarget{QuotientMapsOfCosetSpaces}{}\subsubsection*{{Sequences of coset spaces}}\label{QuotientMapsOfCosetSpaces} Consider $K \hookrightarrow H \hookrightarrow G$ two consecutive group inclusions with their induced coset [[quotient]] [[projections]] \begin{displaymath} \itexarray{ H/K & \longrightarrow& G/K \\ && \downarrow \\ && G/H } \,. \end{displaymath} When $G/K \to G/H$ is a [[Serre fibration]], for instance in the situation of prop. \ref{ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup} (so that this is indeed a [[homotopy fiber sequence]] with respect to the [[classical model structure on topological spaces]]) then it induces the corresponding [[long exact sequence of homotopy groups]] \begin{displaymath} \cdots \to \pi_{n+1}(G/H) \longrightarrow \pi_n(H/K) \longrightarrow \pi_n(G/K) \longrightarrow \pi_n(G/H) \longrightarrow \pi_{n-1}(H/K) \to \cdots \,. \end{displaymath} \begin{example} \label{CofiberSequencesOfCosetsOfOrthogonalGroups}\hypertarget{CofiberSequencesOfCosetsOfOrthogonalGroups}{} Consider a sequence of inclusions of [[orthogonal groups]] of the form \begin{displaymath} O(n) \hookrightarrow O(n+1) \hookrightarrow O(n+k) \,. \end{displaymath} Then by example \ref{nSphereAsCosetSpace} we have that $O(n+1)/O(n) \simeq S^n$ is the [[n-sphere]] and by corollary \ref{QuotientProjectionForCompactLieSubgroupIsPrincipal} the quotient map is a [[Serre fibration]]. Hence there is a [[long exact sequence of homotopy groups]] of the form \begin{displaymath} \cdots \to \pi_q(S^n) \longrightarrow \pi_q(O(n+k)/O(n)) \longrightarrow \pi_q(O(n+k)/O(n+1)) \longrightarrow \pi_{q-1}(S^n) \to \cdots \,. \end{displaymath} Now for $q \lt n$ then $\pi_q(S^n) = 0$ and hence in this range we have [[isomorphisms]] \begin{displaymath} \pi_{\bullet \lt n}(O(n+k)/O(n)) \stackrel{\simeq}{\longrightarrow} \pi_{\bullet \lt n}(O(n+k)/O(n+1)) \,. \end{displaymath} \end{example} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[coset space]] \item [[coadjoint orbit]] \item [[index of a subgroup]] \item [[class equation]] \item [[flag variety]] \item [[Klein geometry]] \item [[WZW model]] \item [[double coset]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item H. Samelson, \emph{Beitrage zur Topologie der Gruppenmannigfaltigkeiten}, Ann. of Math. 2, 42, (1941), 1091 - 1137. \item [[Andrew Gleason]], \emph{Spaces with a compact Lie group of transformations}, Proc. of A.M.S 1, (1950), 35 - 43. \item [[Norman Steenrod]], section I.7 of \emph{The topology of fibre bundles}, Princeton Mathematical Series 14, Princeton Univ. Press, 1951. \item R. Cohen, \emph{Topology of fiber bundles}, Lecture notes (\href{http://math.stanford.edu/~ralph/fiber.pdf}{pdf}) \end{itemize} [[!redirects coset]] [[!redirects cosets]] [[!redirects left coset]] [[!redirects right coset]] [[!redirects left cosets]] [[!redirects right cosets]] [[!redirects coset space]] [[!redirects coset spaces]] \end{document}