+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Group Theory +-- {: .hide} [[!include group theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Given a [[group]] $G$ and a [[subgroup]] $H$, then their _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 _coset space_. This is in particular a [[homogeneous space]], see there for more. ## Definition ### 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 _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 $$ H \times G \underoverset{\mu}{proj_G}\rightrightarrows G $$ 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 $$ G \times H \underoverset{proj_G}{\mu}\rightrightarrows G $$ ### Internal to $Set$ 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 _left coset_. Using [[comprehension]], we can write $$ G/H = \{g H | g \in G\} $$ Similarly there is a coset on the right $H \backslash G$. ### 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 _[[Klein geometry]]_. ### For $\infty$-groups {#ForInfinityGroups} More generally, given an [[(∞,1)-topos]] $\mathbf{H}$ and a [[homomorphism]] of [[∞-group]] objects $H \to G$, 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 $$ \array{ G/H &\longrightarrow& \mathbf{B}H \\ && \downarrow \\ && \mathbf{B}G } \,. $$ See at _[[∞-action]]_ for more on this definition. See at _[[higher Klein geometry]]_ and _[[higher Cartan geometry]]_ for the corresponding concepts of [[higher geometry]]. ## Properties ### 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. ### Topology of the quotient map {#QuotientMaps} +-- {: .num_prop #QuotientProjectionForCompactLieGroupActingFreelyOnManifoldIsPrincipa} ###### Proposition 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]] $$ X \longrightarrow X/G $$ is a $G$-[[principal bundle]] (hence in particular a [[Serre fibration]]). =-- This is originally due to ([Gleason 50](#Gleason50)). See e.g. ([Cohen, theorem 1.3](#Cohen)) +-- {: .num_cor #QuotientProjectionForCompactLieSubgroupIsPrincipal} ###### Corollary For $G$ a [[Lie group]] and $H \subset G$ a [[compact Lie group|compact]] [[subgroup]], then the [[coset]] [[quotient]] [[projection]] $$ G \longrightarrow G/H $$ is an $H$-[[principal bundle]] (hence in particular a [[Serre fibration]]). =-- This is originally due to ([Samelson 41](#Samelson41)). +-- {: .num_prop #ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup} ###### Proposition For $G$ a [[compact Lie group]] and $K \subset H \subset G$ [[closed subspace|closed]] [[subgroups]], then the [[projection]] map $$ p \;\colon\; G/K \longrightarrow G/H $$ is a locally trivial $H/K$-[[fiber bundle]] (hence in particular a [[Serre fibration]]). =-- +-- {: .proof} ###### Proof Observe that the projection map in question is equivalently $$ G \times_H (H/K) \longrightarrow G/H \,, $$ (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}. =-- \begin{prop}\label{CosetSpaceCoprojectionsWithLocalSections} **([[coset space coprojections with local sections]])**\linebreak Let $G$ be a [[topological group]] and $H \subset G$ a [[subgroup]]. Sufficient conditions for the [[coset space]] [[coprojection]] $G \overset{q}{\to} G/H$ to admit [[local sections]], in that there is an [[open cover]] $\underset{i \in I}{\sqcup}U_i \to G/H$ and a [[continuous section]] $\sigma_{\mathcal{U}}$ of the [[pullback]] of $q$ to the cover, $$ \array{ && G_{\vert \mathcal{U}} &\longrightarrow& G \\ & {}^{\mathllap{ \exists \sigma }} \nearrow & \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ \mathllap{ \exists \; } \underset{i \in I}{\sqcup} U_i &=& \underset{i \in I}{\sqcup} U_i &\longrightarrow& G/H \mathrlap{\,,} } $$ include the following: * $G$ is any [[topological group]] and $H$ is a [[compact Lie group]] (in particular for $H$ a [[closed subgroup]] if $G$ itself is a [[compact Lie group]], since [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]). ([Gleason 50, Thm. 4.1](#Gleason50)) or: * The underlying [[topological space]] of $G$ is 1. [[locally compact topological space|locally compact]]; 1. [[separable metric space]]; 1. of [[finite number|finite]] [[dimension of a separable metric space]] (e.g. if $G$ is a [[Lie group]]) and $H \subset G$ is a [[closed subgroup]]. ([Mostert 53, Theorem 3](#Mostert53)) \end{prop} ### As a homotopy fiber \begin{remark}\label{InGeometricHomotopyTheory} 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 $$ \array{ G & \longrightarrow &* \\ \downarrow && \downarrow \\ G/H &\longrightarrow& \mathbf{B}H } \,. $$ 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 reverse [[pasting law]] for [[homotopy pullbacks]] ([here](pasting+law+for+pullbacks#ReversePastingLawForInfinityGroupoids), using that $\ast \to \mathbf{B}H$ is an [effective epimorphism](effective+epimorphism+in+an+infinity1-category) by definition of [[delooping]]) then we get the [[homotopy pullback]] $$ \array{ G/H & \longrightarrow &\mathbf{B}H \\ \downarrow && \downarrow \\ * & \longrightarrow & \mathbf{B}G } $$ which exhibits the coset as the [[homotopy fiber]] of $\mathbf{B}H \to \mathbf{B}G$. See also [SS21, Ex. 3.2.35](#SatiSchreiber21). \end{remark} ## Examples ### $n$-Spheres +-- {: .num_example #nSphereAsCosetSpace} ###### Example The [[n-spheres]] are coset spaces of [[orthogonal groups]]: $$ S^n \simeq O(n+1)/O(n) \,. $$ The odd-dimensional spheres are also coset spaces of [[unitary groups]]: $$ S^{2n+1} \simeq U(n+1)/U(n) $$ =-- +-- {: .proof} ###### 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 $$ \left( \array{ 1 & \vec 0 \\ \vec 0 & A } \right) $$ where $A \in U(n)$. =-- There are also various exceptional realizations of spheres as coset spaces. For instance: [[!include coset space structure on n-spheres -- table]] \linebreak ### Sequences of coset spaces {#QuotientMapsOfCosetSpaces} Consider $K \hookrightarrow H \hookrightarrow G$ two consecutive group inclusions with their induced coset [[quotient]] [[projections]] $$ \array{ H/K & \longrightarrow& G/K \\ && \downarrow \\ && G/H } \,. $$ 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]] $$ \cdots \to \pi_{n+1}(G/H) \longrightarrow \pi_n(H/K) \longrightarrow \pi_n(G/K) \longrightarrow \pi_n(G/H) \longrightarrow \pi_{n-1}(H/K) \to \cdots \,. $$ +-- {: .num_example #CofiberSequencesOfCosetsOfOrthogonalGroups} ###### Example Consider a sequence of inclusions of [[orthogonal groups]] of the form $$ O(n) \hookrightarrow O(n+1) \hookrightarrow O(n+k) \,. $$ 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 $$ \cdots \to \pi_q(S^n) \longrightarrow \pi_q(O(n+k)/O(n)) \longrightarrow \pi_q(O(n+k)/O(n+1)) \longrightarrow \pi_{q-1}(S^n) \to \cdots \,. $$ Now for $q \lt n$ then $\pi_q(S^n) = 0$ and hence in this range we have [[isomorphisms]] $$ \pi_{\bullet \lt n}(O(n+k)/O(n)) \stackrel{\simeq}{\longrightarrow} \pi_{\bullet \lt n}(O(n+k)/O(n+1)) \,. $$ =-- ### Further examples * [[nilmanifolds]] ## Related concepts * [[coset space]] * [[orbit-stabilizer theorem]] * [[coadjoint orbit]] * [[index of a subgroup]] * [[class equation]] * [[flag variety]] * [[Klein geometry]] * [[WZW model]] * [[double coset]] ## References * {#Samelson41} [[Hans Samelson]], _Beiträge zur Topologie der Gruppenmannigfaltigkeiten_, Ann. of Math. 2, 42, (1941), 1091 - 1137. ([jstor:1970463](https://www.jstor.org/stable/1970463), [doi:10.2307/1970463](https://doi.org/10.2307/1970463)) * {#Gleason50} [[Andrew Gleason]], _Spaces with a compact Lie group of transformations_, Proc. of A.M.S 1, (1950), 35 - 43 ([jstor:2032430](https://www.jstor.org/stable/2032430), [doi:10.2307/2032430](https://doi.org/10.2307/2032430)) * {#Steenrod51} [[Norman Steenrod]], Section I.7 of: _The topology of fibre bundles_, Princeton Mathematical Series 14, Princeton Univ. Press, 1951. * {#Mostert53} [[Paul Mostert]], _Local Cross Sections in Locally Compact Groups_, Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp.645-649 ([jstor:2032540](https://www.jstor.org/stable/2032540), [doi:10.2307/2032540](https://doi.org/10.2307/2032540)) * {#Bredon72} [[Glen Bredon]], Section I.4 of: _[[Introduction to compact transformation groups]]_, Academic Press 1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf)) * {#Cohen} R. Cohen, _Topology of fiber bundles_, Lecture notes ([pdf](http://math.stanford.edu/~ralph/fiber.pdf)) On coset spaces with the same [[rational cohomology]] as a [[product space|product]] of [[n-spheres]]: * [[Linus Kramer]], _Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface_, Memoirs of the American Mathematical Society number 752 ([arXiv:math/0109133] (http://arxiv.org/abs/math/0109133), [doi:10.1090/memo/0752](http://dx.doi.org/10.1090/memo/0752), [GoogleBooks](http://books.google.com/books?id=SA8O6ihrDFkC&printsec=frontcover&hl=de&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false)) Discussion in [[(infinity,1)-topos theory|$\infty$-topos theory]]: * {#SatiSchreiber21} [[Hisham Sati]], [[Urs Schreiber]], Ex. 3.2.35 ([p. 104](https://arxiv.org/pdf/2112.13654.pdf#page=104)) of: *[[schreiber:Equivariant Principal infinity-Bundles]]* &lbrack;[arXiv:2112.13654](https://arxiv.org/abs/2112.13654)&rbrack; On [[coset spaces]] ([[homogeneous spaces]]) and their [[Maurer-Cartan forms]] in application to [[first-order formulation of gravity|first-order formulation]] of ([[supergravity|super]]-)[[gravity]]: * [[Leonardo Castellani]], [[L. J. Romans]], [[Nicholas P. Warner]], *Symmetries of coset spaces and Kaluza-Klein supergravity*, Annals of Physics **157** 2 (1984) 394-407 \[<a href="https://doi.org/10.1016/0003-4916(84)90066-6">doi:10.1016/0003-4916(84)90066-6</a>\] * {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], §I.6 in: *[[Supergravity and Superstrings - A Geometric Perspective]]*, World Scientific (1991) &lbrack;[doi:10.1142/0224](https://doi.org/10.1142/0224), toc: [[CDF91-TOC.pdf:file]], ch I.6: [[CastellaniDAuriaFre-ChI6.pdf:file]] * {#Castellani01} [[Leonardo Castellani]], *On G/H geometry and its use in M-theory compactifications*, Annals Phys. **287** (2001) 1-13 &lbrack;[arXiv:hep-th/9912277](https://arxiv.org/abs/hep-th/9912277), [doi:10.1006/aphy.2000.6097](https://doi.org/10.1006/aphy.2000.6097)&rbrack; Further in [[particle physics]]: * Ismaël Ahlouche Lahlali, Josh A. O'Connor: *Coset symmetries and coadjoint orbits* &lbrack;[arXiv:2411.05918](https://arxiv.org/abs/2411.05918)&rbrack; [[!redirects coset]] [[!redirects cosets]] [[!redirects left coset]] [[!redirects right coset]] [[!redirects left cosets]] [[!redirects right cosets]] [[!redirects coset space]] [[!redirects coset spaces]]