sign in sign up
<<
Home , Topological group

Chapter 3

Review of Groups and Actions

3.1 Groups

Definition 3.1. A group is a set G equipped with a bi- nary operation : G G G that associates an element a b G to every· pair⇥ of! elements a, b G,andhaving the· following2 properties: is associative2,hasanidentity element, e G,andeveryelementin· G is invertible (w.r.t. ). More2 explicitly, this means that the following equations· hold for all a, b, c G: 2 (G1) a (b c)=(a b) c.(associativity); · · · · () a e = e a = a.(identity); · · 1 (G3) For every a G,thereissomea G such that 1 12 2 a a = a a = e (inverse). · ·

193 194 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS AgroupG is abelian (or commutative)if a b = b a · · for all a, b G. 2

Observe that a group is never empty, since e G. 2 Given a group, G,foranytwosubsetsR, S G,welet ✓ RS = r s r R, s S . { · | 2 2 } In particular, for any g G,ifR = g ,wewrite 2 { } gS = g s s S { · | 2 } and similarly, if S = g ,wewrite { } Rg = r g r R . { · | 2 }

From now on, we will drop the multiplication sign and write g g for g g . 1 2 1 · 2 3.1. GROUPS 195 Definition 3.2. Given a group, G,asubset,H,ofG is a of G i↵ (1) The identity element, e,ofG also belongs to H (e H); 2 (2) For all h ,h H,wehaveh h H; 1 2 2 1 2 2 1 (3) For all h H,wehaveh H. 2 2

It is easily checked that a subset, H G,isasubgroup ✓ of G i↵ H is nonempty and whenever h1,h2 H,then 1 2 h h H. 1 2 2 If H is a subgroup of G and g G is any element, the sets of the form gH are called left2 of H in G and the sets of the form Hg are called right cosets of H in G. 196 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS The left cosets (resp. right cosets) of H induce an equiv- alence relation, ,definedasfollows:Forallg ,g G, ⇠ 1 2 2 g g i↵ g H = g H 1 ⇠ 2 1 2 (resp. g g i↵ Hg = Hg ). 1 ⇠ 2 1 2 Obviously, is an equivalence relation. Now, it is easy ⇠ 1 to see that g1H = g2H i↵ g2 g1 H,sotheequivalence class of an element g G is the2 gH (resp. Hg). 2 The set of left cosets of H in G (which, in general, is not agroup)isdenotedG/H.The“points”ofG/H are obtained by “collapsing” all the elements in a coset into asingleelement.

The set of right cosets is denoted by H G. \ 3.1. GROUPS 197 It is tempting to define a multiplication operation on left cosets (or right cosets) by setting

(g1H)(g2H)=(g1g2)H, but this operation is not well defined in general, unless the subgroup H possesses a special property.

This property is typical of the kernels of group homomor- phisms.

Definition 3.3. Given any two groups, G, G0,afunction ': G G0 is a homomorphism i↵ ! '(g g )='(g )'(g ), for all g ,g G. 1 2 1 2 1 2 2

Taking g1 = g2 = e (in G), we see that

'(e)=e0, 1 and taking g1 = g and g2 = g ,weseethat 1 1 '(g )='(g) . 198 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS

If ': G G0 and : G0 G00 are group homomor- ! ! phisms, then ': G G00 is also a homomorphism. !

If ': G G0 is a homomorphism of groups and H G ! ✓ and H0 G0 are two , then it is easily checked that ✓

Im H = '(H)= '(g) g H { | 2 } is a subgroup of G0 called the of H by ',and

1 ' (H0)= g G '(g) H0 { 2 | 2 } is a subgroup of G.Inparticular,whenH0 = e0 ,we obtain the ,Ker',of'.Thus, { }

Ker ' = g G '(g)=e0 . { 2 | }

It is immediately verified that ': G G0 is injective i↵ Ker ' = e .(WealsowriteKer' =(0).)! { } 3.1. GROUPS 199 We say that ' is an isomorphism if there is a homomor- phism, : G0 G,sothat ! ' =id and ' =id . G G0

1 In this case, is unique and it is denoted ' .

When ' is an isomorphism we say the the groups G and G0 are isomorphic.WhenG0 = G,agroupisomorphism is called an automorphism.

We claim that H =Ker' satisfies the following prop- erty: gH = Hg, for all g G. ( ) 2 ⇤ First, note that ( )isequivalentto ⇤ 1 gHg = H, for all g G, 2 and the above is equivalent to 1 gHg H, for all g G. ( ) ✓ 2 ⇤⇤ 200 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Definition 3.4. For any group, G,asubgroup, N G,isanormal subgroup of G i↵ ✓ 1 gNg = N, for all g G. 2

This is denoted by N C G.

If N is a of G,theequivalencerela- tion induced by left cosets is the same as the equivalence induced by right cosets.

Furthermore, this equivalence relation, ,isacongru- ⇠ ence,whichmeansthat:Forallg ,g ,g0 ,g0 G, 1 2 1 2 2 (1) If g1N = g10 N and g2N = g20 N,theng1g2N = g10 g20 N, and 1 1 (2) If g1N = g2N,theng1 N = g2 N. As a consequence, we can define a group structure on the set G/ of equivalence classes modulo ,bysetting ⇠ ⇠

(g1N)(g2N)=(g1g2)N. 3.1. GROUPS 201 This group is denoted G/N.Theequivalenceclass,gN, of an element g G is also denoted g.Themap ⇡ : G G/N,givenby2 ! ⇡(g)=g = gN, is clearly a called the canonical projection.

Given a homomorphism of groups, ': G G0,weeasily check that the groups G/Ker ' and Im!' = '(G)are isomorphic. 202 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS 3.2 Group Actions and Homogeneous Spaces, I

If X is a set (usually, some kind of geometric space, for example, the sphere in R3,theupperhalf-plane,etc.),the “” of X are often captured by the action of a group, G,onX.

In fact, if G is a and the action satisfies some simple properties, the set X can be given a manifold structure which makes it a projection (quotient) of G, aso-called“.” 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 203 Definition 3.5. Given a set, X,andagroup,G,aleft action of G on X (for short, an action of G on X)is afunction,': G X X,suchthat ⇥ ! (1) For all g, h G and all x X, 2 2 '(g, '(h, x)) = '(gh, x),

(2) For all x X, 2 '(1,x)=x, where 1 G is the identity element of G. 2

To alleviate the notation, we usually write g x or even gx for '(g, x), in which case, the above axioms· read: (1) For all g, h G and all x X, 2 2 g (h x)=gh x, · · · (2) For all x X, 2 1 x = x. ·

The set X is called a (left) G-set. 204 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS The action ' is faithful or e↵ective i↵for every g,if g x = x for all x X,theng =1; · 2 the action ' is transitive i↵for any two elements x, y X,thereissomeg G so that g x = y. 2 2 ·

Given an action, ': G X X,foreveryg G,we have a function, ' : X ⇥ X!,definedby 2 g ! ' (x)=g x, for all x X. g · 2

Observe that ' has ' 1 as inverse. g g

Therefore, 'g is a bijection of X,i.e.,apermutationof X. 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 205 Moreover, we check immediately that ' ' = ' , g h gh so, the map g ' is a group homomorphism from G 7! g to SX,thegroupofpermutationsofX.

With a slight abuse of notation, this group homomor- phism G S is also denoted '. ! X Conversely, it is easy to see that any group homomor- phism, ': G SX,yieldsagroupaction : G X !X,bysetting · ⇥ ! g x = '(g)(x). · Observe that an action, ', is faithful i↵the group homomorphism, ': G S , is injective. ! X 1 Also, we have g x = y i↵ g y = x. · · 206 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Definition 3.6. Given two G-sets, X and Y ,afunction, f : X Y ,issaidtobeequivariant,oraG-map i↵for all x !X and all g G,wehave 2 2 f(g x)=g f(x). · · Equivalently, if the G-actions are denoted by ': G X X and : G Y Y ,wehavethe following⇥ commutative! diagram⇥ for all!g G: 2 'g X / X f f ✏ ✏ Y / Y. g

Remark: We can also define a right action, : X G X,ofagroupG on a set X,asamap satisfying· ⇥ the! conditions (1) For all g, h G and all x X, 2 2 (x g) h = x gh, · · · (2) For all x X, 2 x 1=x. · 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 207 However, one change is necessary. For every g G,the map ' : X X must be defined as 2 g ! 1 ' (x)=x g , g · in for the map g 'g from G to SX to be a homomorphism (' ' 7!= ' ). g h gh

Conversely, given a homomorphism ': G SX,weget arightaction : X G X by setting! · ⇥ ! 1 x g = '(g )(x). · Every notion defined for left actions is also defined for right actions, in the obvious way. 208 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Here are some examples of (left) group actions.

2 n 1 Example 1:TheunitsphereS (more generally, S ).

n 1 Recall that for any n 1, the (real) unit sphere, S , is the set of points in Rn given by n 1 n 2 2 S = (x1,...,xn) R x + + x =1 . { 2 | 1 ··· n } In particular, S2 is the usual sphere in R3.

Since the group SO(3) = SO(3, R)consistsof(orienta- tion preserving) linear , i.e., linear maps that are distance preserving (and of determinant +1), and ev- ery leaves the origin fixed, we see that any rotation maps S2 into itself. Beware that this would be false if we considered the  group of ane isometries, SE(3), of E3.Forexample, ascrewmotiondoesnot map S2 into itself, even though it is distance preserving, because the origin is translated. 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 209 Thus, we have an action, : SO(3) S2 S2,givenby · ⇥ ! R x = Rx. · The verification that the above is indeed an action is triv- ial. This action is transitive.

Similarly, for any n 1, we get an action, n 1 n 1 : SO(n) S S . · ⇥ ! It is easy to show that this action is transitive.

Analogously, we can define the (complex) unit sphere, n 1 n ⌃ ,asthesetofpointsinC given by

n 1 n ⌃ = (z1,...,zn) C z1z1 + + znzn =1 . { 2 | ··· }

If we write zj = xj + iyj,withxj,yj R,then 2

n 1 2n ⌃ = (x1,...,xn,y1,...,yn) R { 2 | x2 + + x2 + y2 + + y2 =1 . 1 ··· n 1 ··· n } 210 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS

n 1 n Therefore, we can view the complex sphere, ⌃ (in C ), 2n 1 2n as the real sphere, S (in R ).

By analogy with the real case, we can define an action, n 1 n 1 : SU(n) ⌃ ⌃ , · ⇥ !

of the group, SU(n), of linear maps of Cn preserving the hermitian inner product (and the origin, as all linear maps do) and this action is transitive.

n 1 One should not confuse the unit sphere, ⌃ ,withthe  n 1 hypersurface, SC ,givenby n 1 n 2 2 S = (z1,...,zn) C z + + z =1 . C { 2 | 1 ··· n } For instance, one should check that a line, L,through n 1 the origin intersects ⌃ in a circle, whereas it intersects n 1 SC in exactly two points! 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 211 Example 2:Theupperhalf-plane.

The upper half-plane, H,istheopensubsetofR2 con- sisting of all points, (x, y) R2,withy>0. 2 It is convenient to identify H with the set of complex numbers, z C,suchthat z>0. Then, we can define an action, 2 = : SL(2, R) H H, · ⇥ ! as follows: For any z H,foranyA SL(2, R), 2 2 az + b A z = , · cz + d where

ab A = cd ✓ ◆ with ad bc =1. It is easily verified that A z is indeed always well defined and in H when z H.Thisactionistransitive(check· this). 2 212 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Maps of the form az + b z , 7! cz + d where z C and ad bc =1,arecalledM¨obiustrans- formations2 .

Here, a, b, c, d R,butingeneral,weallowa, b, c, d 2 2 C.Actually,thesetransformationsarenotnecessarily defined everywhere on C,forexample,forz = d/c if c =0. 6 To fix this problem, we add a “point at infinity”, ,to 1 C and define M¨obius transformations as functions C C . [{1}! [{1} If c = 0, the M¨obius transformation sends to itself, otherwise, d/c and a/c. 1 7! 1 1 7! 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 213

The space C can be viewed as the plane, R2, extended with[ a{ point1} at infinity.

Using a stereographic projection from the sphere S2 to the plane, (say from the north pole to the equatorial plane), we see that there is a bijection between the sphere, S2, and C . [{1} More precisely, the stereographic projection of the sphere S2 from the north pole, N =(0, 0, 1), to the plane z =0 (extended with the point at infinity, )isgivenby 1 x y x + iy (x, y, z) S2 (0, 0, 1) , = , 2 { }7! 1 z 1 z 1 z ✓ ◆ with (0, 0, 1) . 7! 1 The inverse stereographic projection is given by

2x 2y x2 + y2 1 (x, y) , , , 7! x2 + y2 +1 x2 + y2 +1 x2 + y2 +1 ✓ ◆ with (0, 0, 1). 1 7! 214 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Intuitively, the inverse stereographic projection “wraps” the equatorial plane around the sphere. The space C is known as the Riemann sphere. [{1} We will see shortly that C = S2 is also the complex 1 [{1} ⇠ projective line, CP .

In summary, M¨obius transformations are bijections of the Riemann sphere. It is easy to check that these transfor- mations form a group under composition for all a, b, c, d 2 C,withad bc =1. This is the M¨obiusgroup,denotedM¨ob+.

The M¨obius transformations corresponding to the case a, b, c, d R,withad bc =1formasubgroupof + 2 + M¨ob denoted M¨obR. 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 215 + The map from SL(2, C)toM¨ob that sends A SL(2, C) to the corresponding M¨obius transformation is2 a surjec- tive group homomorphism and one checks easily that its kernel is I,I (where I is the 2 2identitymatrix). { } ⇥ Therefore, the M¨obius group M¨ob+ is isomorphic to the SL(2, C)/ I,I ,denotedPSL(2, C). { } This latter group turns out to be the group of projective 1 transformations of the projective space CP .

+ The same reasoning shows that the subgroup M¨obR is isomorphic to SL(2, R)/ I,I ,denotedPSL(2, R). { } 216 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS

The group SL(2, C)actsonC = S2 the same way [{1} ⇠ that SL(2, R)actsonH,namely:ForanyA SL(2, C), 2 for any z C , 2 [{1} az + b A z = , · cz + d where

ab A = with ad bc =1. cd ✓ ◆ This action is clearly transitive. 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 217 One may recall from complex analysis that the (complex) M¨obius transformation z i z 7! z + i is a biholomorphic isomorphism between the upper half plane, H,andtheopenunitdisk,

D = z C z < 1 . { 2 || | } As a consequence, it is possible to define a transitive ac- tion of SL(2, R)onD.

This can be done in a more direct fashion, using a group isomorphic to SL(2, R), namely, SU(1, 1) (a group of complex matrices), but we don’t want to do this right now. 218 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Example 3:Thesetofn n symmetric, positive, def- inite matrices, SPD(n). ⇥

The group GL(n)=GL(n, R)actsonSPD(n)asfol- lows: For all A GL(n)andallS SPD(n), 2 2

A S = ASA>. ·

It is easily checked that ASA> is in SPD(n)ifS is in SPD(n).

This action is transitive because every SPD , S, can be written as S = AA>,forsomeinvertiblematrix, A (prove this as an exercise). 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 219 n n Example 4:TheprojectivespacesRP and CP .

n The (real) projective space, RP ,isthesetofalllines through the origin in Rn+1,i.e.,thesetofone-dimensional subspaces of Rn+1 (where n 0). Since a one-dimensional subspace, L Rn+1,isspanned ✓ n by any nonzero vector, u L,wecanviewRP as the 2 set of equivalence classes of vectors in Rn+1 0 modulo the equivalence relation, { }

u v i↵ v = u, for some =0 R. ⇠ 6 2

In terms of this definition, there is a projection, n+1 n pr : R 0 RP , { }! given by pr(u)=[u] ,theequivalenceclassofu modulo . ⇠ ⇠ Write [u]forthelinedefinedbythenonzerovector,u. 220 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS

Since every line, L,inRn+1 intersects the sphere Sn in n two antipodal points, we can view RP as the quotient of the sphere Sn by identification of antipodal points.

n We define an action of SO(n +1)onRP as follows: For any line, L =[u], for any R SO(n +1), 2 R L =[Ru]. · Since R is linear, the line [Ru]iswelldefined,i.e.,does not depend on the choice of u L.Itisclearthatthis action is transitive. 2

n The (complex) projective space, CP ,isdefinedanalo- gously as the set of all lines through the origin in Cn+1, i.e., the set of one-dimensional subspaces of Cn+1 (where n 0). 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 221 n This time, we can view CP as the set of equivalence classes of vectors in Cn+1 0 modulo the equivalence relation, { }

u v i↵ v = u, for some =0 C. ⇠ 6 2

We have the pro jection, n+1 n pr : C 0 CP , { }! given by pr(u)=[u] ,theequivalenceclassofu modulo . ⇠ ⇠ Again, write [u]forthelinedefinedbythenonzerovector, u.

n n n Remark: Algebraic geometers write PR for RP and PC n (or even Pn)forCP . 222 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS

Recall that ⌃n Cn+1,theunitsphereinCn+1,isdefined by ✓

n n+1 ⌃ = (z1,...,zn+1) C z1z1+ +zn+1zn+1 =1 . { 2 | ··· }

For any line, L =[u], where u Cn+1 is a nonzero vector, 2 n+1 writing u =(u1,...,un+1), a point z C belongs to 2 L i↵ z = (u1,...,un+1), for some C. 2 Therefore, the intersection, L ⌃n,ofthelineL and the sphere ⌃n is given by \

n n+1 L ⌃ = (u1,...,un+1) C \ { 2 | C,(u1u1 + + un+1un+1)=1 , 2 ··· } i.e.,

n n+1 L ⌃ = (u1,...,un+1) C C, \ 2 2 ⇢ 1 = . 2 2 | | u1 + + un+1 ) | | ··· | | p 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 223 Thus, we see that there is a bijection between L ⌃n and the circle, S1,i.e.,geometrically,L ⌃n is a circle.\ \ Moreover, since any line, L,throughtheoriginisdeter- mined by just one other point, we see that for any two lines L1 and L2 through the origin, L = L i↵(L ⌃n) (L ⌃n)= . 1 6 2 1 \ \ 2 \ ; However, ⌃n is the sphere S2n+1 in R2n+2.

n It follows that CP is the quotient of S2n+1 by the equiv- alence relation, ,definedsuchthat ⇠ y z i↵ y, z L ⌃n, ⇠ 2 \ for some line, L,throughtheorigin.

Therefore, we can write 2n+1 1 n S /S ⇠= CP . 224 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS The case n =1isparticularlyinteresting,asitturnsout that

3 1 2 S /S ⇠= S .

This is the famous Hopf fibration.Toshowthis,proceed as follows: As 3 1 2 2 2 S = ⌃ = (z,z0) C z + z0 =1 , ⇠ { 2 || | | | } define a map, HF: S3 S2,by ! 2 2 HF((z,z0)) = (2zz , z z0 ). 0 | | | |

We leave as a homework exercise to prove that this map has range S2 and that

HF((z1,z10 )) = HF((z2,z20 )) i↵

(z ,z0 )=(z ,z0 ), for some with =1. 1 1 2 2 | | 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 225 In other words, for any point, p S2,theinverseimage, 1 2 3 HF (p)(alsocalledfibre over p), is a circle on S .

Consequently, S3 can be viewed as the union of a family of disjoint circles. This is the Hopf fibration.

It is possible to visualize the Hopf fibration using the stereographic projection from S3 onto R3.Thisisabeau- tiful and puzzling picture. For example, see Berger [2].

1 Therefore, HF induces a bijection from CP to S2,andit is a .

n We define an action of SU(n +1)onCP as follows: For any line, L =[u], for any R SU(n +1), 2 R L =[Ru]. · Again, this action is well defined and it is transitive. 226 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Example 5:Anespaces.

If E is any (real) vector space and X is any set, a transi- tive and faithful action, : E X X,oftheadditive group of E on X makes ·X into⇥ an!ane space.

The intuition is that the members of E are translations.

Those familiar with ane spaces as in Gallier [16] (Chap- ter 2) or Berger [2] will point out that if X is an ane space, then, not only is the action of E on X transitive, but more is true:

For any two points, a, b E,thereisaunique vector, u E,suchthatu a = 2b. 2 · 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 227 By the way, the action of E on X is usually considered to be a right action and is written additively, so u a is written a + u (the result of translating a by u). ·

Thus, it would seem that we have to require more of our action.

However, this is not necessary because E (under ) is abelian.

Proposition 3.1. If G is an acting on a set X and the action : G X X is transitive and faithful, then for any two· elements⇥ ! x, y X, there is a unique g G so that g x = y (the action2 is simply transitive).2 ·

More examples will be considered later. 228 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS The subset of group elements that leave some given ele- ment x X fixed plays an important role. 2

Definition 3.7. Given an action, : G X X,ofa group G on a set X,foranyx X·,thegroup⇥ !G (also 2 x denoted StabG(x)), called the stabilizer of x or isotropy group at x is given by G = g G g x = x . x { 2 | · }

It is easy to verify that Gx is indeed a subgroup of G.

In general, Gx is not anormalsubgroup.

Observe that 1 Gg x = gGxg , · for all g G and all x X. 2 2 Therefore, the stabilizers of x and g x are conjugate of each other. · 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 229 When the action of G on X is transitive, for any fixed x G,thesetX is a quotient (as a set, not as group) of 2 G by Gx.

Proposition 3.2. If : G X X is a transitive action of a group G on· a set⇥X, for! every fixed x X, the surjection, ⇡ : G X, given by 2 ! ⇡(g)=g x · induces a bijection ⇡ : G/G X, x ! where Gx is the stabilizer of x.

The map ⇡ : G X (corresponding to a fixed x X) is sometimes called! a projection of G onto X. 2

Proposition 3.2 shows that for every y X,thesubset 1 2 ⇡ (y), (called the fibre above y)isequaltosomecoset gGx of G,andthusisinbijectionwiththegroupGx itself. 230 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS

We can think of G as a moving family of fibres, Gx, parametrized by X.

This point of view of viewing a space as a moving family of simpler spaces is typical in (algebraic) geometry, and underlies the notion of (principal) fibre bundle.

Note that if the action : G X X is transitive, then · ⇥ ! the stabilizers Gx and Gy of any two elements x, y X are isomorphic, as they as conjugates. 2

Thus, in this case, it is enough to compute one of these stabilizers for a “convenient” x.

Definition 3.8. Aset,X,issaidtobeahomogeneous space if there is a transitive action, : G X X,of some group, G,onX. · ⇥ !

We see that all the spaces of Example 1–5 are homoge- neous spaces. 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 231 Another example that will play an important role when we deal with Lie groups is the situation where we have a group, G,asubgroup,H,ofG (not necessarily normal) and where X = G/H,thesetofleftcosetsofG modulo H.

The group G acts on G/H by left multiplication: a (gH)=(ag)H, · where a, g G.Thisactionisclearlytransitiveandone 2 1 checks that the stabilizer of gH is gHg .

If G is a and H is a closed subgroup of G (see later for an explanation), it turns out that G/H is Hausdor↵ .

If G is a Lie group, we obtain a manifold. 232 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Even if G and X are topological spaces and the action,  : G X X,iscontinuous,thespaceG/Gx under the· quotient⇥ ! topology is, in general, not homeomorphic to X.

We will give later sucient conditions that insure that X is indeed a or even a manifold.

In particular, X will be a manifold when G is a Lie group.

In general, an action : G X X is not transitive on X,butforeveryx ·X,itistransitiveontheset⇥ ! 2 O(x)=G x = g x g G . · { · | 2 } Such a set is called the orbit of x.Theorbitsarethe equivalence classes of the following equivalence relation: 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 233 Definition 3.9. Given an action, : G X X,of some group, G,onX,theequivalencerelation,· ⇥ !,onX is defined so that, for all x, y X, ⇠ 2 x y i↵ y = g x, for some g G. ⇠ · 2 For every x X,theequivalenceclassofx is the orbit of x,denoted2O(x)orG x,with · O(x)=G x = g x g G . · { · | 2 } The set of orbits is denoted X/G.

We warn the reader that some authors use the notation G X for the the set of orbits G x,becausetheseorbits can\ be considered as right orbits,· by analogy with right cosets Hg of a subgroup H of G.

The orbit space, X/G,isobtainedfromX by an identi- fication (or merging) process: For every orbit, all points in that orbit are merged into a single point. 234 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS For example, if X = S2 and G is the group consisting of the restrictions of the two linear maps I and I of R3 to S2 (where ( I)(x)= x), then 2 2 X/G = S / I, I = RP . { } ⇠

More generally, if Sn is the n-sphere in Rn+1,thenwe have a bijection between the orbit space Sn/ I, I and n { } RP :

n n S / I, I = RP . { } ⇠ Many manifolds can be obtained in this fashion, including the , the Klein bottle, the M¨obius band, etc.

Since the action of G is transitive on O(x), by Proposition 3.2, we see that for every x X,wehaveabijection 2

O(x) ⇠= G/Gx. 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 235 As a corollary, if both X and G are finite, for any set A X of representatives from every orbit, we have the orbit✓ formula:

X = [G: G ]= G / G . | | a | | | a| a A a A X2 X2 Even if a , : G X X,isnottransi- tive, when X is a manifold,· we⇥ can! consider the set of orbits, X/G,andiftheactionofG on X satisfies certain conditions, X/G is actually a manifold.

Spaces arising in this fashion are often called orbifolds.

In summary, we see that manifolds arise in at least two ways from a group action:

(1) As homogeneous spaces, G/Gx,iftheactionistran- sitive. (2) As orbifolds, X/G. Of course, in both cases, the action must satisfy some additional properties. 236 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Let us now determine some stabilizers for the actions of Examples 1–4, and for more examples of homogeneous spaces.

(a) Consider the action n 1 n 1 : SO(n) S S , · ⇥ ! n 1 of SO(n)onthesphereS (n 1) defined in Example 1. Since this action is transitive, we can determine the n 1 stabilizer of any convenient element of S ,say e1 =(1, 0,...,0).

In order for any R SO(n)toleavee fixed, the first 2 1 column of R must be e1,soR is an orthogonal matrix of the form

1 U R = , with det(S)=1. 0 S ✓ ◆ As the rows of R must be unit vector, we see that U =0 and S SO(n 1). 2 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 237

Therefore, the stabilizer of e1 is isomorphic to SO(n 1), and we deduce the bijection

n 1 SO(n)/SO(n 1) = S . ⇠

Strictly speaking, SO(n 1) is not a subgroup of  SO(n)andinallrigor,weshouldconsiderthesub- group, SO(n 1), of SO(n)consistingofallmatricesof the form 10 g , with det(S)=1 0 S ✓ ◆ and write

n 1 SO(n)/SO(n 1) = S . ⇠ However, it is commong practice to identify SO(n 1) with SO(n 1). 1 Whengn =2,asSO(1) = 1 ,wefindthatSO(2) ⇠= S , acircle,afactthatwealreadyknew.{ } 238 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS

2 When n =3,wefindthatSO(3)/SO(2) ⇠= S . This says that SO(3) is somehow the result of gluing circles to the surface of a sphere (in R3), in such a way that these circles do not intersect. This is hard to visualize!

n 1 Asimilarargumentforthecomplexunitsphere,⌃ , shows that

n 1 2n 1 SU(n)/SU(n 1) = ⌃ = S . ⇠ ⇠ Again, we identify SU(n 1) with a subgroup of SU(n), as in the real case. In particular, when n =2,as SU(1) = 1 ,wefindthat { } 3 SU(2) ⇠= S , i.e., the group SU(2) is topologically the sphere S3!

Actually, this is not surprising if we remember that SU(2) is in fact the group of unit quaternions. 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 239 (b) We saw in Example 2 that the action : SL(2, R) H H · ⇥ ! of the group SL(2, R)ontheupperhalfplaneistransi- tive. Let us find out what the stabilizer of z = i is.

We should have ai + b = i, ci + d that is, ai + b = c + di,i.e., (d a)i = b + c. Since a, b, c, d are real, we must have d = a and b = c. Moreover, ad bc =1,sowegeta2 + b2 =1. We conclude that a matrix in SL(2, R)fixesi i↵it is of the form a b , with a2 + b2 =1. ba ✓ ◆ 240 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Clearly, these are the rotation matrices in SO(2) and so, the stabilizer of i is SO(2). We conclude that

SL(2, R)/SO(2) ⇠= H.

This time, we can view SL(2, R)astheresultofgluing circles to the upper half plane. This is not so easy to visualize.

There is a better way to visualize the topology of SL(2, R) by making it act on the open disk, D.Wewillreturnto this action in a little while. 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 241

2 Now, consider the action of SL(2, C)onC ⇠= S . As it is transitive, let us find the stabilizer[ of{z1=0.We} must have b =0, d and as ad bc =1,wemusthaveb =0andad =1.

Thus, the stabilizer of 0 is the subgroup, SL(2, C)0,of SL(2, C)consistingofallmatricesoftheform a 0 1 , where a C 0 and c C. ca 2 { } 2 ✓ ◆ We get

2 SL(2, C)/SL(2, C)0 = C = S , ⇠ [{1} ⇠ but this is not very illuminating. 242 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS (c) In Example 3, we considered the action : GL(n) SPD(n) SPD(n) · ⇥ ! of GL(n)onSPD(n), the set of symmetric positive def- inite matrices.

As this action is transitive, let us find the stabilizer of I. For any A GL(n), the matrix A stabilizes I i↵ 2

AIA> = AA> = I.

Therefore, the stabilizer of I is O(n)andwefindthat

GL(n)/O(n)=SPD(n).

Observe that if GL+(n)denotesthesubgroupofGL(n) consisting of all matrices with a strictly positive determi- nant, then we have an action : GL+(n) SPD(n) SPD(n). · ⇥ !

This action is transtive and we find that the stabilizer of I is SO(n); consequently, we get

GL+(n)/SO(n)=SPD(n). 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 243 (d) In Example 4, we considered the action n n : SO(n +1) RP RP · ⇥ ! n of SO(n+1)onthe(real)projectivespace, RP .Asthis action is transitive, let us find the stabilizer of the line, L =[e1], where e1 =(1, 0,...,0).

We find that the stabilizer of L =[e1]isisomorphicto the group O(n)andso,

n SO(n +1)/O(n) ⇠= RP .

Strictly speaking, O(n)isnotasubgroupofSO(n+1),  so the above equation does not make sense. We should write

n SO(n +1)/O(n) ⇠= RP ,

where O(n)isthesubgroupofe SO(n +1)consistingof all matrices of the form ↵ 0e , with S O(n),↵= 1, det(S)=↵. 0 S 2 ± ✓ ◆ However, the common practice is to write O(n)instead of O(n).

e 244 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS

3 We should mention that RP and SO(3) are homeomor- phic spaces. This is shown using the quaternions, for example, see Gallier [16], Chapter 8.

Asimilarargumentappliestotheaction n n : SU(n +1) CP CP · ⇥ ! n of SU(n +1)onthe(complex)projectivespace,CP . We find that

n SU(n +1)/U(n) ⇠= CP .

Again, the above is a bit sloppy as U(n)isnotasubgroup of SU(n+1). To be rigorous, we should use the subgroup, U(n), consisting of all matrices of the form ↵ 0 e , with S U(n), ↵ =1, det(S)=↵. 0 S 2 | | ✓ ◆ The common practice is to write U(n)insteadofU(n). In particular, when n =1,wefindthat e 1 SU(2)/U(1) ⇠= CP .

3 1 But, we know that SU(2) ⇠= S and, clearly, U(1) ⇠= S . 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 245

3 1 1 So, again, we find that S /S ⇠= CP (but we know, 3 1 2 1 more, namely, S /S ⇠= S ⇠= CP .) (e) We now consider a generalization of projective spaces (real and complex). First, consider the real case.

Given any n 1, for any k,with0 k n,letG(k, n)   be the set of all linear k-dimensional subspaces of Rn (also called k-planes).

Any k-dimensional subspace, U,ofR is spanned by k n linearly independent vectors, u1,...,uk,inR ;writeU = span(u1,...,uk).

We can define an action, : O(n) G(k, n) G(k, n) · ⇥ ! as follows: For any R O(n), for any 2 U =span(u1,...,uk), let

R U =span(Ru ,...,Ru ). · 1 k We have to check that the above is well defined but this is not hard. 246 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS It is also easy to see that this action is transitive.

Thus, it is enough to find the stabilizer of any k-plane.

We can show that the stabilizer of U is isomorphic to O(k) O(n k)andwefindthat ⇥ O(n)/(O(k) O(n k)) = G(k, n). ⇥ ⇠ It turns out that this makes G(k, n)intoasmoothman- ifold of dimension k(n k)calledaGrassmannian. The restriction of the action of O(n)onG(k, n)toSO(n) yields an action : SO(n) G(k, n) G(k, n) · ⇥ ! of SO(n)onG(k, n). 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 247 Then, it is easy to see that the stabilizer of the subspace U is isomorphic to the subgroup S(O(k) O(n k)) of SO(n)consistingoftherotationsoftheform⇥ S 0 R = , 0 T ✓ ◆ with S O(k), T O(n k)anddet(S)det(T )=1. 2 2 Thus, we also have

SO(n)/S(O(k) O(n k)) = G(k, n). ⇥ ⇠

n If we recall the projection pr : Rn+1 0 RP ,by n { }! definition, a k-plane in RP is the image under pr of any (k +1)-planeinRn+1.

n So, for example, a line in RP is the image of a 2-plane n in Rn+1,andahyperplaneinRP is the image of a hy- perplane in Rn+1.

The advantage of this point of view is that the k-planes n in RP are arbitrary, i.e., they do not have to go through “the origin” (which does not make sense, anyway!). 248 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Then, we see that we can interpret the Grassmannian G(k +1,n +1)asaspaceof“parameters”forthek- n planes in RP .Forexample,G(2,n+1)parametrizes n the lines in RP .

In this viewpoint, G(k +1,n +1)isusuallydenoted G(k, n).

It can be proved (using some ) that G(k, n) (n) 1 can be embedded in RP k .

Much more is true. For example, G(k, n)isaprojective variety, which means that it can be defined as a subset of (n) 1 RP k equal to the zero locus of a set of homogeneous equations.

There is even a set of quadratic equations, known as the Pl¨uckerequations,definingG(k, n). 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 249 In particular, when n =4andk =2,wehaveG(2, 4) 5 ✓ RP ,andG(2, 4) is defined by a single equation of degree 2.

The Grassmannian G(2, 4) = G(1, 3) is known as the 5 Klein quadric.ThishypersurfaceinRP parametrizes 3 the lines in RP .Itplayanimportantroleincomputer vision.

Complex Grassmannians are defined in a similar way, by replacing R by C throughout.

The complex Grassmannian, GC(k, n), is a complex man- ifold as well as a real manifold and we have

U(n)/(U(k) U(n k)) = G (k, n). ⇥ ⇠ C 250 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS As in the case of the real Grassmannians, the action of

U(n)onGC(k, n)yieldsanactionofSU(n)onGC(k, n), and we get

SU(n)/S(U(k) U(n k)) = G (k, n), ⇥ ⇠ C where S(U(k) U(n k)) is the subgroup of SU(n) consisting of all⇥ matricesR SU(n)oftheform 2 S 0 R = , 0 T ✓ ◆ with S U(k), T U(n k)anddet(S)det(T )=1. 2 2 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 251 (f) Closely related to Grassmannians are the Stiefel man- ifolds.

We begin with the real case.

For any n 1andanyk with 1 k n,letS(k, n)be the set of all orthonormal k-frames ,thatis,of k-tuples n of orthonormal vectors (u1,...,uk)withui R . 2 n 1 Obviously, S(1,n)=S and S(n, n)=O(n), so as- sume k n 1.  There is a natural action : SO(n) S(k, n) S(k, n) · ⇥ ! of SO(n)onS(k, n)givenby

R (u ,...,u )=(Ru ,...,Ru ). · 1 k 1 k This action is transitive 252 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Let us find the stabilizer of the orthonormal k-frame (e1,...,ek)consistingofthefirstcanonicalk-basis vec- tors of Rn.

AmatrixR SO(n)stabilizes(e1,...,ek)i↵itisofthe form 2

I 0 R = k 0 S ✓ ◆ where S SO(n k). 2 Therefore,

SO(n)/SO(n k) = S(k, n). ⇠ This makes S(k, n)asmoothmanifoldofdimension

k(k +1) k(k 1) nk = k(n k)+ . 2 2 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 253 Remark: It should be noted that we can define another type of Stiefel manifolds, denoted by V (k, n), using lin- early independent k-tuples (u1,...,uk)thatdonotnec- essarily form an orthonormal system.

In this case, there is an action : GL(n, R) V (k, n) V (k, n) · ⇥ ! and the stabilizer H of the first k canonical basis vec- tors (e1,...,ek)isaclosedsubgroupofGL(n, R), but it doesn’t have a simple description,

We get an isomorphism

V (k, n) ⇠= GL(n, R)/H.

The version of the Stiefel manifold S(k, n)usingorthonor- mal frames is sometimes denoted by V 0(k, n) (Milnor and 0 n Stashe↵[27] use the notation Vk (R )).

Beware that the notation is not standardized. Certain authors use V (k, n)forwhatwedenotebyS(k, n)! 254 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Complex Stiefel manifolds are defined in a similar way by replacing R by C and O(n)byU(n).

For 1 k n 1, the complex Stiefel manifold SC(k, n) is isomorphic  to the quotient

SU(n)/SU(n k) = S (k, n). ⇠ C

2n 1 If k =1,wehaveSC(1,n)=S ,andifk = n,we have SC(n, n)=U(n).

The Grassmannians can also be viewed as quotient spaces of the Stiefel manifolds.

Every k-frame (u1,...,uk)canberepresentedbyan n k matrix Y over the canonical basis of Rn,andsuch amatrix⇥ Y satisfies the equation

Y >Y = I. 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 255 We have a right action : S(k, n) O(k) S(k, n) · ⇥ ! given by

Y R = YR, · for any R O(k). 2 However, this action is not transitive (unless k =1), but the orbit space S(k, n)/O(k)isisomorphictothe Grassmannian G(k, n), so we can write

G(k, n) ⇠= S(k, n)/O(k).

Similarly, the complex Grassmannian is isomorphic to the orbit space SC(k, n)/U(k):

GC(k, n) ⇠= SC(k, n)/U(k). 256 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS We now return to case (b) to give a better picture of SL(2, R). Instead of having SL(2, R)actontheupper half plane we define an action of SL(2, R)ontheopen unit disk, D.

Technically, it is easier to consider the group, SU(1, 1), which is isomorphic to SL(2, R), and to make SU(1, 1) act on D.

The group SU(1, 1) is the group of 2 2complexmatrices of the form ⇥

ab , with aa bb =1. b a ✓ ◆ 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 257 The reader should check that if we let 1 i g = , 1 i ✓ ◆ then the map from SL(2, R)toSU(1, 1) given by

1 A gAg 7! is an isomorphism.

Observe that the M¨obius transformation associated with g is z i z , 7! z +1 which is the holomorphic isomorphism mapping H to D mentionned earlier!

Now, we can define a bijection between SU(1, 1) and S1 D given by ⇥ ab (a/ a ,b/a). b a 7! | | ✓ ◆ 258 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS

We conclude that SL(2, R) ⇠= SU(1, 1) is topologically an open solid torus (i.e., with the surface of the torus removed).

It is possible to further classify the elements of SL(2, R) into three categories and to have geometric interpreta- tions of these as certain regions of the torus.

For details, the reader should consult Carter, Segal and Macdonald [9] or Duistermatt and Kolk [14] (Chapter 1, Section 1.2).

The group SU(1, 1) acts on D by interpreting any matrix in SU(1, 1) as a M¨obius tranformation, i.e.,

ab az + b z . b a 7! 7! bz + a ✓ ◆ ✓ ◆ The reader should check that these transformations pre- serve D. 3.2. GROUP ACTIONS AND HOMOGENEOUS SPACES, I 259 Both the upper half-plane and the open disk are mod- els of Lobachevsky’s non- (where the parallel postulate fails).

They are also models of hyperbolic spaces (Riemannian manifolds with constant negative curvature, see Gallot, Hulin and Lafontaine [17], Chapter III).

According to Dubrovin, Fomenko, and Novikov [13] (Chap- ter 2, Section 13.2), the open disk model is due to Poincar´e and the upper half-plane model to Klein, although Poincar´e was the first to realize that the upper half-plane is a hy- perbolic space. 260 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS 3.3 Topological Groups

Since Lie groups are topological groups (and manifolds), it is useful to gather a few basic facts about topological groups.

Definition 3.10. Aset,G,isatopological group i↵ (a) G is a Hausdor↵topological space; (b) G is a group (with identity 1); (c) Multiplication, : G G G,andtheinverseop- · ⇥ !1 eration, G G: g g ,arecontinuous,where G G has! the product7! topology. ⇥

It is easy to see that the two requirements of condition (c) are equivalent to 1 (c0) The map G G G:(g, h) gh is continuous. ⇥ ! 7! 3.3. TOPOLOGICAL GROUPS 261 Given a topological group G,foreverya G we de- fine left translation as the map, L : G G2,suchthat a ! La(b)=ab,forallb G,andright translation as the map, R : G G,suchthat2 R (b)=ba,forallb G. a ! a 2

Observe that L 1 is the inverse of L and similarly, R 1 a a a is the inverse of Ra.Asmultiplicationiscontinuous,we see that La and Ra are continuous.

Moreover, since they have a continuous inverse, they are .

As a consequence, if U is an open subset of G,thensois gU = L (U)(resp.Ug = R U), for all g G. g g 2 Therefore, the topology of a topological group (i.e., its family of open sets) is determined by the knowledge of the open subsets containing the identity, 1. 262 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS

1 1 Given any subset, S G,letS = s s S ;let S0 = 1 and Sn+1 =✓SnS,foralln { 0. | 2 } { } Property (c) of Definition 3.10 has the following useful consequences:

Proposition 3.3. If G is a topological group and U is any open subset containing 1, then there is some 1 open subset, V U, with 1 V , so that V = V and V 2 U. Furthermore,✓ V 2 U. ✓ ✓

1 Asubset,U,containing1suchthatU = U ,iscalled symmetric.

Using Proposition 3.3, we can give a very convenient char- acterization of the Hausdor↵separation property in a topological group. 3.3. TOPOLOGICAL GROUPS 263 Proposition 3.4. If G is a topological group, then the following properties are equivalent: (1) G is Hausdor↵; (2) The set 1 is closed; { } (3) The set g is closed, for every g G. { } 2

If H is a subgroup of G (not necessarily normal), we can form the set of left cosets, G/H and we have the projection, p: G G/H,wherep(g)=gH = g. ! If G is a topological group, then G/H can be given the quotient topology,whereasubsetU G/H is open i↵ 1 ✓ p (U)isopeninG.

With this topology, p is continuous.

The trouble is that G/H is not necessarily Hausdor↵. However, we can neatly characterize when this happens. 264 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Proposition 3.5. If G is a topological group and H is a subgroup of G then the following properties hold: (1) The map p: G G/H is an open map, which means that p(V )!is open in G/H whenever V is open in G. (2) The space G/H is Hausdor↵i↵ H is closed in G. (3) If H is open, then H is closed and G/H has the discrete topology (every subset is open).

(4) The subgroup H is open i↵ 1 H (i.e., there is some open subset, U, so that 2 1 U H). 2 ✓

Proposition 3.6. If G is a connected topological group, then G is generated by any symmetric neigh- borhood, V , of 1. In fact, G = V n. n 1 [ 3.3. TOPOLOGICAL GROUPS 265 Asubgroup,H,ofatopologicalgroupG is discrete i↵the induced topology on H is discrete, i.e., for every h H, there is some open subset, U,ofG so that U H =2h . \ { }

Proposition 3.7. If G is a topological group and H is discrete subgroup of G, then H is closed.

Proposition 3.8. If G is a topological group and H is any subgroup of G, then the , H, of H is a subgroup of G.

Proposition 3.9. Let G be a topological group and H be any subgroup of G.IfH and G/H are connected, then G is connected. 266 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Proposition 3.10. Let G be a topological group and let V be any connected symmetric open subset con- taining 1. Then, if G0 is the connected component of the identity, we have n G0 = V n 1 [ and G0 is a normal subgroup of G. Moreover, the group G/G0 is discrete.

Atopologicalspace,X is locally compact i↵for every point p X,thereisacompactneighborhood,C of p, i.e., there2 is a compact, C,andanopen,U,with p U C. 2 ✓ For example, manifolds are locally compact. 3.3. TOPOLOGICAL GROUPS 267 Proposition 3.11. Let G be a topological group and assume that G is connected and locally compact. Then, G is countable at infinity, which means that G is the union of a countable family of compact subsets. In fact, if V is any symmetric compact neighborhood of 1, then G = V n. n 1 [

Definition 3.11. Let G be a topological group and let X be a topological space. An action ': G X X is continuous (and G acts continuously on X⇥)ifthemap! ' is continuous.

If an action ': G X X is continuous, then each ⇥ ! map 'g : X X is a homeomorphism of X (recall that ' (x)=g x!,forallx X). g · 2 268 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Under some mild assumptions on G and X,thequotient space G/Gx is homeomorphic to X.Forexample,this happens if X is a Baire space.

Recall that a Baire space X is a topological space with the property that if F i 1 is any countable family of { } closed sets Fi such that each Fi has empty interior, then i 1 Fi also has empty interior. ByS complementation, this is equivalent to the fact that for every countable family of open sets Ui such that each Ui is dense in X (i.e., U i = X), then i 1 Ui is also dense in X. T

Remark: AsubsetA X is rare if its closure A has empty interior. A subset✓ Y X is meager if it is a countable union of rare sets. ✓

Then, it is immediately verified that a space X is a Baire space i↵every nonempty open subset of X is not meager. 3.3. TOPOLOGICAL GROUPS 269 The following theorem shows that there are plenty of Baire spaces:

Theorem 3.12. (Baire) (1) Every locally compact topological space is a Baire space. (2) Every is a Baire space.

Theorem 3.13. Let G be a topological group which is locally compact and countable at infinity, X a Haus- dor↵topological space which is a Baire space, and as- sume that G acts transitively and continuously on X. Then, for any x X, the map ': G/Gx X is a homeomorphism. 2 !

By Theorem 3.12, we get the following important corol- lary: 270 CHAPTER 3. REVIEW OF GROUPS AND GROUP ACTIONS Theorem 3.14. Let G be a topological group which is locally compact and countable at infinity, X a locally compact Hausdor↵topological space and assume that G acts transitively and continuously on X. Then, for any x X, the map ': G/Gx X is a homeomor- phism.2 !

Remark: If a topological group acts continuously and transitively on a Hausdor↵topological space, then for ev- ery x X,thestabilizer,Gx,isaclosedsubgroupof G. 2

This is because, as the action is continuous, the projection ⇡ : G X : g g x is continuous, and !1 7! · G = ⇡ ( x ), with x closed. x { } { }