Chapter 18 Manifolds Arising from Group Actions 18.1 Proper Maps We saw in Chapter 3 that many manifolds arise from a group action. The scenario is that we have a smooth action ': G M M of a Lie group G acting on a manifold M (recall⇥ that! an action ' is smooth if it is a smooth map). If G acts transitively on M,thenforanypointx M, 2 if Gx is the stabilizer of x,thenweknowfromProposi- tion ?? that G/Gx is di↵eomorphic to M and that the projection ⇡ : G G/G is a submersion. ! x 823 824 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS If the action is not transitive, then we consider the orbit space M/G of orbits G x. · However, in general, M/G is not even Hausdor↵. It is thus desirable to look for sufficient conditions that ensure that M/G is Hausdor↵. Asufficientconditioncanbegivenusingthenotionofa proper map. Before we go any further, let us observe that the case where our action is transitive is subsumed by the more general situation of an orbit space. Indeed, if our action is transitive, for any x M,weknow 2 that the stabilizer H = Gx of x is a closed subgroup of G. 18.1. PROPER MAPS 825 Then, we can consider the right action G H G of H on G given by ⇥ ! g h = gh, g G, h H. · 2 2 The orbits of this (right) action are precisely the left cosets gH of H. Therefore, the set of left cosets G/H (the homogeneous space induced by the action : G M M)istheset of orbits of the right action G· H⇥ G!. ⇥ ! Observe that we have a transitive left action of G on the space G/H of left cosets, given by g g H = g g H. 1 · 2 1 2 The stabilizer of 1H is obviously H itself. 826 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Thus, we recover the original transitive left action of G on M = G/H. Now, it turns out that an action of the form G H G, where H is a closed subgroup of a Lie group⇥ G!,isa special case of a free and proper action M G G,in which case the orbit space M/G is a manifold,⇥ ! and the projection ⇡ : G M/G is a submersion. ! Let us now define proper maps. Definition 18.1. If X and Y are two Hausdor↵topo- logical spaces,1 acontinuousmap': X Y is proper i↵for every topological space Z,themap! ' id: X Z Y Z is a closed map (recall that f is⇥ a closed⇥ map! i↵the⇥ image of any closed set by f is a closed set). If we let Z be a one-point space, we see that a proper map is closed. 1It is not necessary to assume that X and Y are Hausdor↵but, if X and/or Y are not Hausdor↵, we have to replace “compact” by “quasi-compact.” We have no need for this extra generality. 18.1. PROPER MAPS 827 If F is any closed subset of X,thentherestrictionof' to F is proper (see Bourbaki, General Topology [9], Chapter 1, Section 10). The following result can be shown (see Bourbaki, General Topology [9], Chapter 1, Section 10): Proposition 18.1. A continuous map ': X Y is 1 ! proper i↵ ' is closed and if '− (y) is compact for every y Y . 2 1 If ' is proper, it is easy to show that '− (K)iscompact in X whenever K is compact in Y . Moreover, if Y is also locally compact, then we have the following result (see Bourbaki, General Topology [9], Chapter 1, Section 10). 828 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Proposition 18.2. If Y is locally compact, a map 1 ': X Y is a proper map i↵ '− (K) is compact in X whenever! K is compact in Y In particular, this is true if Y is a manifold since manifolds are locally compact. This explains why Lee [35] (Chapter 9) takes the property stated in Proposition 18.2 as the definition of a proper map (because he only deals with manifolds). Finally, we can define proper actions. 18.2. PROPER AND FREE ACTIONS 829 18.2 Proper and Free Actions Definition 18.2. Given a Hausdor↵topological group G and a topological space M,aleftaction : G M M is proper if it is continuous and if the map· ⇥ ! ✓ : G M M M, (g, x) (g x, x) ⇥ ! ⇥ 7! · is proper. If H is a closed subgroup of G and if : G M M is aproperaction,thentherestrictionofthisactionto· ⇥ ! H is also proper. If we let M = G,thenG acts on itself by left translation, and the map ✓ : G G G G given by ✓(g, x)= (gx, x)isahomeomorphism,soitisproper.⇥ ! ⇥ 830 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS It follows that the action : H G G of a closed subgroup H of G on G (given· by⇥ (h, g)! hg)isproper. 7! The same is true for the right action of H on G. As desired, proper actions yield Hausdor↵orbit spaces. Proposition 18.3. If the action : G M M is proper (where G is Hausdor↵), then· the⇥ orbit! space M/G is Hausdor↵. Furthermore, M is also Haus- dor↵. We also have the following properties (see Bourbaki, Gen- eral Topology [9], Chapter 3, Section 4). 18.2. PROPER AND FREE ACTIONS 831 Proposition 18.4. Let : G M M be a proper action, with G Hausdor↵·. For⇥ any x! M, let G x be 2 · the orbit of x and let Gx be the stabilizer of x. Then: (a) The map g g x is a proper map from G to M. 7! · (b) Gx is compact. (c) The canonical map from G/Gx to G x is a home- omorphism. · (d) The orbit G x is closed in M. · If G is locally compact, we have the following character- ization of being proper (see Bourbaki, General Topology [9], Chapter 3, Section 4). Proposition 18.5. If G and M are Hausdor↵and G is locally compact, then the action : G M M is proper i↵for all x, y M, there exist· some⇥ open! sets, V and V in M, with2x V and y V , so that the x y 2 x 2 y closure K of the set K = g G Vx g Vy = , is compact in G. { 2 | · \ 6 ;} 832 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS In particular, if G has the discrete topology, the above condition holds i↵the sets g G Vx g Vy = are finite. { 2 | · \ 6 ;} Also, if G is compact, then K is automatically compact, so every compact group acts properly. If M is locally compact, we have the following character- ization of being proper (see Bourbaki, General Topology [9], Chapter 3, Section 4). Proposition 18.6. Let : G M M be a con- tinuous action, with G and· M⇥ Hausdor! ↵. For any compact subset K of M we have: (a) The set G = g G g K K = is closed. K { 2 | · \ 6 ;} (b) If M is locally compact, then the action is proper i↵ GK is compact for every compact subset K of M. 18.2. PROPER AND FREE ACTIONS 833 In the special case where G is discrete (and M is locally compact), condition (b) says that the action is proper i↵ GK is finite. Remark: If G is a Hausdor↵topological group and if H is a subgroup of G,thenitcanbeshownthattheaction of G on G/H ((g1,g2H) g1g2H)isproperi↵H is compact in G. 7! Definition 18.3. An action : G M M is free if for all g G and all x M,if· g =1then⇥ !g x = x. 2 2 6 · 6 An equivalent way to state that an action : G M M · ⇥ ! is free is as follows. For every g G,letLg : M M be the di↵eomorphism of M given2 by ! L (x)=g x, x M. g · 2 Then, the action : G M M is free i↵for all g G, if g =1thenL has· no⇥ fixed! point. 2 6 g 834 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Another equivalent statement is that for every x M, the stabilizer G of x is reduced to the trivial group2 1 . x { } If H is a subgroup of G,obviouslyH acts freely on G (by multiplication on the left or on the right). Before stating the main results of this section, observe that in the definition of a fibre bundle (Definition ??), the local trivialization maps are of the form 1 ' : ⇡− (U ) U F, ↵ ↵ ! ↵ ⇥ where the fibre F appears on the right. In particular, for a principal fibre bundle ⇠,thefibreF is equal to the structure group G,andthisisthereason why G acts on the right on the total space E of ⇠ (see Proposition ??). 18.2. PROPER AND FREE ACTIONS 835 To be more precise, we could call such bundles right bun- dles. We can also define left bundles as bundles whose local trivialization maps are of the form 1 ' : ⇡− (U ) F U . ↵ ↵ ! ⇥ ↵ Then, if ⇠ is a left principal bundle, the group G acts on E on the left. Duistermaat and Kolk [19] address this issue at the end of their Appendix B, and prove the theorem stated below (Chapter 1, Section 11). Beware that in Duistermaat and Kolk [19], this theorem is stated for left bundles,whichisadeparturefromthe prevalent custom of dealing with right bundles. 836 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS However, the weaker version that does not mention princi- pal bundles is usually stated for left actions; for instance, in Lee [35] (Chapter 9, Theorem 9.16).