Transitive actions of locally compact groups on locally contractible spaces
aa r X i v : . [ m a t h . G R ] J u l Transitive actions of locally compact groupson locally contractible spaces
Karl H. Hofmann and Linus Kramer
Abstract
Suppose that X = G/K is the quotient of a locally compact group by a closed subgroup.If X is locally contractible and connected, we prove that X is a manifold. If the G -actionis faithful, then G is a Lie group. In 1974 J. Szenthe stated the following result in [34].
Let a σ -compact locally compact group G ,with compact quotient G/G ◦ , act as a transitive and faithful transformation group on a locallycontractible space X . Then X is a manifold and G is a Lie group. This result, which maybe viewed as a solution of Hilbert’s 5th problem for transformation groups, has been widelyused since then. However, it was discovered in 2011 that Szenthe’s proof contains a seriousgap. In the present paper we close this gap, proving Szenthe’s statement in a different way.Independently and simultaneously, this result was also proved by A.A. George Michael [12] andby S. Antonyan and T. Dobrowolski [3]. The last section of our paper contains some morecomments on the history of this problem. Our main results are as follows.
Theorem A
Let G be a compact group and let K ⊆ G be a closed subgroup. Suppose that thehomogeneous space X = G/K contains a nonempty open subset V ⊆ X which is contractiblein X . Then X is a closed manifold. If N = T { gKg − | g ∈ G } denotes the kernel of the G -action, then G/N is a compact Lie group acting transitively on X . For the case of a locally compact group, we need a stronger topological assumption on the cosetspace X . Theorem B
Let G be a locally compact group and let K ⊆ G be a closed subgroup. Supposethat the homogeneous space X = G/K is locally contractible. Then X is a manifold. If X isconnected or if G/G ◦ is compact, and if N = T { gKg − | g ∈ G } denotes the kernel of theaction, then G/N is a Lie group acting transitively on X . In the course of the proof, we need the following extension of Iwasawa’s Local Splitting The-orem, which may be interesting in its own right. A local version of this result was proved byGluˇskov [13]. This result is also proved in [19] Theorem 4.1 in a different way.1 heorem C
Let G be a locally compact group and let U ⊆ G be a neighborhood of the identity.Then there exist a compact subgroup N ⊆ U , a simply connected Lie group L and an openhomomorphism ϕ : N × L ✲ G with discrete kernel, such that ϕ ( n,
1) = n for all n ∈ N . A variation of the theme of this article appears in the third edition of [20] (2013) in Sections10.72 to 10.93.
Conventions and Terminology
All maps and group homomorphisms are assumed to becontinuous and all spaces and groups are assumed to be Hausdorff unless stated otherwise.Topological countability assumptions will be stated explicitly whenever they are used. By a Liegroup we mean a locally compact group G which is a smooth manifold, such that multiplicationand inversion are smooth maps, without any further topological countability assumptions.The identity component of a topological group G is denoted by G ◦ . This is always a closednormal subgroup. We denote by N ( G ) the collection of all closed normal subgroups N ✂ G with the property that G/N is a Lie group, and we note that G ∈ N ( G ).For subgroups P, Q ⊆ G we putCen P ( Q ) = { p ∈ P | pq = qp for all q ∈ Q } Cen( Q ) =Cen Q ( Q )Nor P ( Q ) = { p ∈ P | pQp − = Q } [ P, Q ] = h [ p, q ] | p ∈ P and q ∈ Q i The unit interval is denoted by [0 ,
1] = { t ∈ R | ≤ t ≤ } . For a map h : X × [0 , ✲ Y we write h t ( x ) = h ( x, t ). The projection map of a cartesian product X × Y is denoted bypr X : ( x, y ) x , and similarly pr Y : ( x, y ) y . Acknowledgment
We thank S. Antonyan, S. Morris, and the referee for their helpful com-ments. Also, we are grateful to R. McCallum for his careful reading of the manuscript.
We begin by collecting some results which we shall need in the proof of Theorem A. Unlessstated otherwise, homotopies are not required to preserve base points. Recall that a map E ✲ B is called a fibration if it has the homotopy lifting property for every space X . Thismeans that for every commutative diagram X × { } ✲ EX × [0 , ❄ ∩ h ✲ ˜ h ................................ ✲ B, ❄ the dotted lift ˜ h exists. The following result is related to the notion of irreducibility in [9] p. 394and in [27]. 2 .1 Lemma Let E be a space with the property that every homotopy equivalence E ≃ ✲ E is surjective. Suppose that p : E ✲ B is a surjective fibration. Then also every homotopyequivalence B ≃ ✲ B is surjective. If p is homotopic to a map p ′ : E ✲ B , then p ′ is alsosurjective. Proof.
First we note the following. If ξ : B ✲ B is a homotopy equivalence withhomotopy inverse η , then ξ ◦ η is, by definition, homotopic to id B . In order to show thesurjectivity of such a map ξ , it suffices therefore to prove the surjectivity of every map whichis homotopic to the identity id B .Suppose that h : B × [0 , ✲ B is a map with h = id B . Then the map h ′ : E × [0 , ✲ B with h ′ t ( x ) = h t ( p ( x )) is a homotopy between p = h ′ and p ′ = h ′ . We now showthat p ′ is surjective. Since E is a fibration, there exists a lift ˜ h ′ : E × [0 , ✲ E of h ′ , with˜ h ′ = id E . EE × [0 , h ′ ✲ ˜ h ′ ................................ ✲ B ❄ By our assumptions on E , the map ˜ h ′ : E ✲ E is surjective. It follows that B = p (˜ h ′ ( E )) = p ′ ( E ) = h ( B ). ✷ We now recall several results about the structure of compact groups.
Let G be a compact group. Thenevery neighborhood V of the identity contains a closed normal subgroup N ✂ G such that G/N is a compact Lie group. The set N ( G ) consisting of all closed normal subgroups N ✂ G suchthat G/N is a Lie group is a filter basis converging to the identity.
Proof.
See Theorem 9.1 and 2.43 in [20]. ✷ Let G be a compact group and N ✂ G a closed normal subgroup. If N and G/N are Lie groups, then G is a Lie group as well. Proof.
We show that G has no small subgroups, see [20] Theorem 2.40. Let W ⊆ G/N bea neighborhood of the identity which contains no nontrivial subgroup and let V be its preimagein G under the projection G ✲ G/N . Let U ⊆ G be a neighborhood of the identity suchthat U ∩ N does not contain a nontrivial subgroup of N . Then U ∩ V contains no nontrivialsubgroup of G . ✷ Suppose that G is a compactgroup and that N ✂ G is a closed normal subgroup. If G/N is connected, then there exists aclosed connected subgroup M ✂ G ◦ with G = M N and [ M, N ] = { } , and such that M ∩ N istotally disconnected, Proof.
See Theorem 9.77 of [20]. ✷ Suppose that X is a set and that x ∈ X . Let Sym( X ) denote the group of all permutationsof X . Suppose that M ⊆ Sym( X ) is a transitive subgroup, with x -stabilizer H = M x . Thenwe may identify X with the quotient M/H in an M -equivariant way, thereby identifying x with H ∈ M/H . Let N = Nor M ( H ) ⊆ M . Then N acts (from the left) on M/H via( n, mH ) mHn − = mn − H . In this way we obtain a homomorphism α ✲ H ⊂ ✲ N α ✲ Sym( X )with α ( n )( mH ) = mn − H . The kernel of α is H . Obviously, the factor group N/H centralizesin this action the group M .We claim that α maps N onto the centralizer C = Cen Sym( X ) ( M ). Suppose that c ∈ C .By transitivity of M , there exists n ∈ M with n − ( x ) = c ( x ). For h ∈ H we have then hn − ( x ) = hc ( x ) = ch ( x ) = c ( x ) = n − ( x ), whence n ∈ N . For m ∈ M we have cm ( x ) = mc ( x ) = mn − ( x ). The right-hand side is precisely the N -action defined above.We need, however, a topological version of this fact which is somewhat stronger than [29] p. 73. Suppose that G is a compact group, that K ⊆ G is a closed subgroup such that K contains no nontrivial normal subgroup of G . Suppose that M ⊆ G is a closed subgroupsuch that G = KM . Then the compact group Cen G ( M ) injects continuously into the compactgroup Nor M ( M ∩ K ) /M ∩ K . Proof.
Let N = Nor M ( M ∩ K ) and H = M ∩ K . By 2.5 we may view Cen G ( M ) as asubgroup of the abstract group C = Cen Sym(
G/K ) ( M ). By 2.5 we have an injective abstractgroup homomorphism β : Cen G ( M ) ✲ N/H . The graph β = { ( c, nH ) ∈ Cen G ( M ) × N/H | nc ∈ K } of β is closed and thus β is continuous, see for example [9] XI.2.7. ✷ Let M be a compact connected group. If there exists a closed totally discon-nected normal subgroup D ⊆ M such that M/D is a Lie group, then there exist simple simplyconnected compact Lie groups S , . . . , S r and a compact connected finite dimensional abeliangroup A and a central surjective homomorphism A × S × · · · × S r ✲ M with a totally disconnected kernel. Proof.
By [20] Proposition 9.47, the Lie algebra of M is isomorphic to the Lie algebra of M/D and hence has a finite dimension. By [20] Theorem 9.52, M has the properties claimedabove. ✷ Let G be a compact group and let P, Q ⊆ G be closedsubgroups, with P ⊆ Q . Then the natural map G/P ✲ G/Q is a fibration.
Proof.
Suppose we are given a commutative diagram X × { } ✲ G/PX × [0 , ❄ ∩ ✲ ............................ ✲ G/Q. ❄ We have to show the existence of the dotted map. To this end we consider the poset P consistingof pairs ( ϕ, N ), where N ✂ G is a closed normal subgroup and ϕ : X × [0 , ✲ G/P N is amap that fits into the commutative diagram X × { } ✲ G/P ✲ G/P NX × [0 , ❄ ∩ ✲ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ G/Q ❄ ✲
G/QN. ❄ We put ( ϕ, N ) ≥ ( ψ, M ) if N ⊆ M and if the diagram G/P ✲ G/P N ✲ G/P MX × [0 , ✲ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ G/Q ❄ ✲
G/QN ❄ ✲
G/QM. ❄ commutes. We claim that this partial order is inductive. Suppose that T ⊆ P is a linearlyordered subset. Let L = T { N | ( N, ϕ ) ∈ T } ✂ G . We have natural maps Y ( N,ϕ ) ∈T G/P N ✛ α G/P LX × [0 , δ ✲ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ γ ✲ Y ( N,ϕ ) ∈T G/QN ❄ ✛ β G/QL ❄ Now α and β are injective and hence homeomorphisms onto their respective images. Moreover,the image of γ is contained in the image of α and the image of δ is contained in the image of β .5hus we can fill in the dotted map and obtain an upper bound ( L, ψ ) of T . By Zorn’s Lemma, P has maximal elements.Suppose that N ✂ G is a compact normal subgroup and that G/N is a Lie group. Thereforethe canonical map
G/P N ✲ G/QN is a locally trivial fiber bundle, see [35], Theorem 3.58,and hence a fibration, see [9], Theorem 4.2. Thus there exists a map ϕ : X × [0 , ✲ G/P N such that (
N, ϕ ) is contained in P . By Theorem 2.2, there exist arbitrarily small compactnormal subgroups N ✂ G such that G/N is a Lie group. It follows that the maximal elementsin P are of the form ( { } , ϕ ), and these elements solve the initial lifting problem. ✷ A more general result than Theorem 2.8 can be found in [32] Theorem 15. As a commentarywe mention the following corollary (which will not be used here).
Let G be a compact group and let P ⊆ Q ⊆ G be closed subgroups. Thenthere is a long exact sequence for the homotopy groups ✲ π k ( Q/P ) ✲ π k ( G/P ) ✲ π k ( G/Q ) ✲ . ✷ Similarly, the Leray-Serre Spectral Sequence may be applied to such a fibration (because ev-ery fibration is a Serre fibration). For the next lemma we remark that ˇCech cohomologyand Alexander-Spanier cohomology agree for compact spaces. See also [22] App. II 3.15 andApp. III 2.11.
Suppose that a compact totally disconnected group D acts on a compact space X . Then the orbit space map p : X ✲ D \ X induces an injection in ˇCech cohomology withrational coefficients, ˇ H ∗ ( X ; Q ) ✛ p ∗ ˇ H ∗ ( D \ X ; Q ) ✛ . Proof.
We have by Theorem 2.2 that D = lim ←−{ D/E | E ∈ N ( D ) } and the groups D/E are finite (in other words, D is a profinite group). It follows that X = lim ←−{ E \ X | E ∈ N ( D ) } . For
E, F ∈ N ( D ) with F ⊆ E we have that E/F is finite and thusˇ H ∗ ( F \ X ; Q ) ✛ ˇ H ∗ ( E \ X ; Q )is injective, see [7] Theorem III.7.2. Projective limits of compact spaces commute with ˇCechcohomology, see [10] X.3. The claim follows now. ✷ The following result is also proved in [18] Corollary 1.9.
Let G be a compact group, let P ⊆ G be a closedsubgroup, and let ξ : G/P ✲ G/P be a homotopy equivalence. Then ξ is surjective. roof. As we noted in the proof of Lemma 2.1, we have to show that for every map h : G/P × [0 , ✲ G/P with h = id G/P , the map h is surjective. Let C denote the class ofall pairs ( G, P ) of compact groups where this holds. Our goal is to show that C is the class ofall compact group pairs. Claim 1. If G is a compact connected group, then ( G, is in C . Suppose that this is false.Then there exists a map h : G × [0 , ✲ G with h = id G , and h : G ✲ G is not surjective.By Theorem 2.2, there exists a closed normal subgroup N ✂ G such that G/N is a compactconnected Lie group, say of dimension r , and an element g ∈ G such that gN ∩ h ( G ) = ∅ .By Theorem 2.4, there exists a compact connected subgroup M ⊆ G such that G = M N ,and D = M ∩ N is totally disconnected. Then L = M/D ∼ = G/N is a compact connected r -dimensional Lie group. We note that r >
0, since otherwise we would have G = N . Then wehave a commutative diagram M ⊂ ✲ G ✛ j ⊃ h ( G ) ✛ h GL ❄ ∼ = ✲ G/N ❄ ✛
G/N − { gN } . ❄ Because of the homotopy h ≃ h = id G , the restriction map ˇ H ∗ ( G ; Q ) j ∗ ✲ ˇ H ∗ ( h ( G ); Q ) isinjective. We have therefore in r -dimensional cohomology a commutative diagramˇ H r ( M ; Q ) ✛ ˇ H r ( G ; Q ) j ∗ ✲ ˇ H r ( h ( G ); Q )ˇ H r ( L ; Q )(1) ✻ ✛ ∼ = ˇ H r ( G/N ; Q )(2) ✻ ✲ ˇ H r ( G/N − { gN } ; Q ) . ✻ The map (1) is injective by Lemma 2.10. Thus (2) is also injective. The map j ∗ is injective by theprevious remark. The compact connected Lie group L is Q -orientable, whence ˇ H r ( L ; Q ) ∼ = Q .On the other hand, ˇ H r ( G/N − { gN } ; Q ) = 0. This is a contradiction. Claim 2. Suppose that G is a compact group. Then ( G, is in C . Suppose that a ∈ G is notin the image of h : G ✲ G . Then h ′ t ( g ) = a − h t ( ag ) is a map from the identity component G ◦ of G to itself with h ′ = id G ◦ , and 1 is not in the image of h ′ , contradicting Claim 1. Claim 3. If ( G, is in C and if P ⊆ G is closed, then ( G, P ) is in C . This follows from Claim 2and Lemma 2.1, since G ✲ G/P is a fibration by Theorem 2.8. ✷ Let V be a subset of a topological space X . We say that V is contractible in X if there isa map f : V × [0 , ✲ X , ( v, t ) f t ( v ), such that f ( v ) = v for all v ∈ V and f isa constant map from V into X . In the case that V is open, we note that the image of f in X is a path-connected set with nonempty interior. We call a topological space X piecewisecontractible if it satisfies the following condition.7 PC)
There is a cover of X by nonempty open subsets which are contractible in X .If X admits a transitive group of homeomorphisms, then (PC) is obviously equivalent to thecondition that there is some nonempty open set V ⊆ X which is contractible in X . A product space X × Y is piecewise contractible of and only if X and Y arepiecewise contractible. Proof. If V ⊆ X and W ⊆ Y are open subsets and if f : V × [0 , ✲ X and g : W × [0 , ✲ Y contract V and W in X and Y , respectively, then the map h : V × W × [0 , ✲ X × Y , with h t ( v, w ) = ( f t ( v ) , g t ( w )) contracts V × W in X × Y . In this way weobtain the required open cover of X × Y .Conversely, if U ⊆ X × Y is an open subset containing the point ( x, y ), and if h : U × [0 , ✲ X × Y contracts U in X × Y , then there is an open neighborhood V ⊆ X of x suchthat V × { y } ⊆ U . Then f t ( v ) = pr X ( h t ( v, y )) contracts V in X . ✷ We now prove three preparatory lemmas in order to obtain Theorem A.
Let G be a compact group and let K ⊆ G be a closed subgroup. If X = G/K is piecewise contractible, then the subgroup G ◦ K is open in G , the quotient space G ◦ K/K ∼ = G ◦ /G ◦ ∩ K is piecewise contractible, and there is a G ◦ -equivariant homeomorphism G/K ∼ = ( G ◦ K/K ) × D for some finite set D . Proof.
The space
G/G ◦ K is compact and totally disconnected, see [20] Prop. 10.32 andthe following remark. Thus the free right action of the compact group G ◦ K on G has a totallydisconnected orbit space. Now the result [20] Theorem 10.35 on the Existence of Global CrossSections applies and shows that G is homeomorphic to G ◦ K × D with a totally disconnectedcompact space D (homeomorphic to G/G ◦ K ) in such a fashion that the action of G ◦ K is bymultiplication on the first factor. In other words, we have a G ◦ K -equivariant homeomorphism G ∼ = G ◦ K × ( G/G ◦ K ), and therefore a G ◦ -equivariant homeomorphism G/K ∼ = ( G ◦ K/K ) × ( G/G ◦ K ) . Now
G/K is piecewise contractible. Hence by Lemma 3.1, the totally disconnected compacthomogeneous space D ∼ = G/G ◦ K is piecewise contractible. This implies that each point of D is open, and hence D is finite. So G ◦ K is open in G and G ◦ K/K is open in
G/K . Lemma 3.1implies also that G ◦ K/K is piecewise contractible. ✷ Let G be a compact connected group and let K ⊆ G be a closed subgroup. If X = G/K is piecewise contractible, then there exists a closed normal connected subgroup M ✂ G with G = KM (i.e. M acts transitively on X ), and M has a closed central totallydisconnected subgroup D ✂ M such that M/D is a compact connected Lie group.
Proof.
Let V ⊆ G/K be an open neighborhood of the coset K ∈ G/K which is contractiblein X by a map f : V × [0 , ✲ G/K . By Theorem 2.2 there exists a closed normal subgroup8 ∈ N ( G ) with N K/K ⊆ V . Let M ✂ G be a complement of N as in Theorem 2.4. Since G = M N , the group N acts transitively on G/KM , and we have by Theorem 2.8 a surjectivefibration
N K/K ∼ = N/K ∩ N p ✲ N/ ( KM ) ∩ N ∼ = G/KM.
We define f ′ : ( N/K ∩ N ) × [0 , ✲ G/KM by f ′ t ( n ( K ∩ N )) = f t ( nK ) KM . Then f ′ = p and f ′ is constant. It follows from Lemma 2.1 that G = KM . Let D = M ∩ N . Then M/D ∼ = G/N is a compact connected Lie group. Since [
M, N ] = { } , we have that D is central in M . ✷ The proof of the following lemma is partially adapted from [23] Prop. 3.5.
Let
G, K, X, M be as in Lemma 3.3. If G acts faithfully on X , then M is acompact Lie group. Proof.
We put L = M ∩ K and we identify X with the quotient space M/L . It follows fromTheorem 2.7 that there is a compact connected semisimple Lie group S , a compact connectedfinite dimensional abelian group A , and a surjective homomorphism q : A × S ✲ M with atotally disconnected kernel E . Passing to a quotient, we can also assume that E intersects thefactors { } × S and A × { } trivially. We want to show that A is a Lie group.Since M acts faithfully and transitively on X , the central subgroup q ( A × { } ) intersectsthe stabilizer L trivially and hence A acts freely on M/L . There is an open neighborhood V ⊆ X of L ∈ M/L and a map f : V × [0 , ✲ X that contracts V in X . By Theorem 2.2, there is a closed subgroup B ⊆ A such that A/B ∼ = T m is a compact torus of finite dimension m , and such that the B -orbit of L ∈ M/L is containedin V . From the action of A × S on M/L we have by Theorem 2.8 a fibration A × S ✲ M/L
We note that B acts freely on M/L . Hence we may identify B with the B -orbit of L ∈ M/L .Then f gives us a map g : B × [0 , ✲ M/L , with g ( b ) = bL and g constant. We now liftthis to ˜ g : B × [0 , ✲ A × S , such that ˜ g ( b ) = b for all b ∈ B . We define h : B × [0 , ✲ A by h t ( b ) = pr A (˜ g t ( b )). Now there is a covering homomorphism ϕ : B × R m ✲ A with discretekernel, with ϕ ( b,
0) = b for all b ∈ B , see [21] Thm. 13.17 and 13.20. We lift h to a map˜ h : B × [0 , ✲ B × R m . The composite B × [0 , ˜ h ✲ B × R m pr B ✲ B is a homotopy between id B and a constant map. Thus B is contractible and hence by 2.11trivial. ✷ Now we have collected all ingredients for the proof of Theorem A.
Suppose that G is a compact group and that K is a closed subgroup. Assumealso that the action of G on the homogeneous space X = G/K is faithful, i.e. that K contains o nontrivial normal subgroup of G . If X is piecewise contractible, then G is a compact Liegroup and X is a closed, but not necessarily connected manifold. Proof.
By Lemma 3.2 we have a G ◦ -equivariant homeomorphism G/K ∼ = ( G ◦ /G ◦ ∩ K ) × D ,for some finite set D , and G ◦ /G ◦ ∩ K is piecewise contractible. By Lemma 3.4 there existsa closed normal connected Lie group M ✂ G ◦ acting transitively on G ◦ /G ◦ ∩ K . The group G ◦ decomposes by Theorem 2.4 as G ◦ = N M , where N ✂ G is a closed normal subgroupthat centralizes M . Since G ◦ acts faithfully and transitively on G ◦ /G ◦ ∩ K , it follows fromLemma 2.6 that N is isomorphic to a closed subgroup of the Lie group Nor M ( M ∩ K ) /M ∩ K .By Lemma 2.3, the group G ◦ is a compact connected Lie group. Now we want to show the samefor the group G ◦ K . The group Cen G ( G ◦ ) is normal in G and has therefore trivial intersectionwith K , because the action is faithful. Thus Cen K ( G ◦ ) = { } . Therefore K injects into thecompact Lie group Aut( G ◦ ). Now both G ◦ and G ◦ K/G ◦ ∼ = K/K ∩ G ◦ are Lie groups and thus G ◦ K is also a Lie group by Lemma 2.3. Finally, G ◦ K is open in G , hence G is also a Lie group. ✷ Let G be a compact group and K a closed subgroup. If X = G/K is piecewisecontractible, then
G/K is a closed, but not necessarily connected manifold. The quotient
G/N ,where N = T { gKg − | g ∈ G } is a compact Lie group that acts faithfully and transitively on X . ✷ Theorem 3.5 and Corollary 3.6 yield Theorem A in the introduction. For the sake of complete-ness, we restate the result in terms of transformation groups.
Let X be a compact locally contractible space. Suppose that a compact group G acts as a transitive transformation group on X , via a continuous map G × X ✲ X. Then X is a closed manifold. If the G -action is faithful, then G is a compact Lie group. Proof.
Let K denote the stabilizer of a point x ∈ X . Since G is compact, the naturalcontinuous map G/K ✲ X is a homeomorphism. ✷ The following result was proved by Iwasawa in [25], p. 547, Theorem 11.
Let G be a locally compact con-nected group. Then G has arbitrarily small neighborhoods which are of the form N C suchthat N is a compact normal subgroup and C is an open n-cell which is a local Lie groupcommuting elementwise with N , such that ( n, c ) nc is a homeomorphism N × C ✲ N C . ✷ loc.cit. that G is a projective limit of Lie groups. However, in the processof settling Hilbert’s Fifth Problem (see [28], p. 184), Yamabe showed that every locally compactgroup has an open subgroup which is a projective limit of Lie groups (see [28], p. 175). Wenow extend Iwasawa’s Splitting Theorem to not necessarily connected locally compact groups.This result is essentially Gluˇskov’s Theorem A in [13]. It is also proved in [19] Theorem 4.1 ina different way. We begin with two lemmas. Let A and B compact connected abelian groups. Suppose that we are givencontinuous homomorphisms R m ϕ ✲ B ✛ p A . If p is surjective, then the lifting problem A R m ϕ ✲ ˜ ϕ ................................. ✲ Bp ❄ has a solution ˜ ϕ . Proof.
We dualize the diagram. The Pontrjagin duals b A and b B are discrete torsion freeabelian groups and b p is injective. Moreover, c R m ∼ = R m . The dual problem b A c R m ✛ b ϕ b ˜ ϕ ✛ ................................. b B b p ✻ clearly has a solution b ˜ ϕ (for example by passing to the divisible hulls of b A and b B , which are Q -vector spaces). Note that we do not have to worry about continuity, since both b A and b B arediscrete groups. Now we dualize the solution b ˜ ϕ of this problem. ✷ Let L be a simply connected Lie group and let N be a compact group. Let α : L ✲ Aut( N ) be a homomorphism, and consider the semidirect product N ⋊ α L . If L centralizes under this map the identity component N ◦ , then there is an isomorphism ϕ : N ⋊ α L ∼ = ✲ N × L which restricts to the identity on N × . Proof.
The α -image of L is contained in the identity component Aut( N ) ◦ , because L is con-nected. On the other hand, we have a natural injective map N/ Cen( N ) = Inn( N ) ⊂ ✲ Aut( N ).Under this map, Aut( N ) ◦ ∼ = N ◦ / Cen( N ) ∩ N ◦ , see [25] p. 514, Theorem 1 ′ or [20] Theorem 9.82.The subgroup of Aut( N ) ◦ that centralizes N ◦ is therefore isomorphic to Cen( N ◦ ) / Cen( N ) ∩ N ◦ .Thus the L -action on N is given by a map L ✲ (Cen( N ◦ ) / Cen( N ) ∩ N ◦ ) ◦ . The target groupof this map is a quotient of the compact connected abelian group Cen( N ◦ ) ◦ . Since this groupis in particular abelian, we end up with a map L ab ✲ L/ [ L, L ] ψ ✲ (Cen( N ◦ ) / Cen( N ) ∩ N ◦ ) ◦ .11ow L is simply connected and thus L/ [ L, L ] ∼ = R m , for some m ≥
0. By Lemma 4.2, thereexists a lift ˜ ψ : L/ [ L, L ] ✲ Cen( N ◦ ) ◦ . Now we consider the composite β = j ◦ ˜ ψ ◦ ab, L ab ✲ L/ [ L, L ] ˜ ψ ✲ Cen( N ◦ ) ◦ j ✲ N. where j ( x ) = x − . Then we have β ( ℓ ) − nβ ( ℓ ) = α ( ℓ )( n )for all n ∈ N and ℓ ∈ L . Now N × L ✲ N ⋊ L, ϕ ( n, ℓ ) = ( nβ ( ℓ ) , ℓ )is the desired isomorphism. ✷ The following result, which is Theorem C in our introduction, is a global version of Gluˇskov’sTheorem A in [13].
Let G be a locally compact group. Then for every identity neighborhood U there is a compact subgroup N contained in U , a simply connected Lie group L , and an openand continuous homomorphism ϕ : N × L ✲ G with discrete kernel such that ϕ ( n,
1) = n for all n ∈ N . Proof.
We divide the proof into several steps.
Claim 1. The result holds if G is connected. We apply Iwasawa’s Local Splitting Theorem 4.1.The fact that C is a local Lie group on an open n -cell means that there is a Lie group L , an n -cell identity neighborhood W ⊆ L , and a homeomorphism γ : W ✲ C for which x, y, xy ∈ W implies γ ( xy ) = γ ( x ) γ ( y ). We may assume L to be simply connected. Then γ extends to aunique homomorphism of topological groups γ : L ✲ G , see [20], Corollary A2.26 and A2.27.Since C is in the centralizer of N , so is the subgroup γ ( L ) generated by C . Hence the map ϕ : N × L ✲ G, ϕ ( n, ℓ ) = nγ ( ℓ ) , is a continuous homomorphism which maps N × W homeomorphically onto the identity neigh-borhood N C of G . Thus ker ϕ is discrete and ϕ is locally open and hence open. Clearly ϕ ( n,
1) = n . The assertion follows in this special case. Claim 2. The result holds if
G/G ◦ is compact. Then every identity neighborhood contains acompact normal subgroup P such that G/P is a Lie group, see [28] Ch. 4.6, p. 175. Let U ⊆ G be an identity neighborhood. By Theorem 4.1, the identity component G ◦ has a relatively openidentity neighborhood QC ∼ = Q × C with a compact normal subgroup Q ✂ G ◦ contained in U and an open n -cell local Lie group C . We may assume that the n -cell C contains no subgroupbesides { } . Let ψ : Q × L ✲ G ◦ be the surjective homomorphism guaranteed by Claim 1of the proof, and put γ : L ✲ G, γ ( ℓ ) = ψ (1 , ℓ ) . N ( G ) be the filter basis of compact normal subgroups P ✂ G such that G/P is a Liegroup. Since the filter basis N ( G ) converges to 1, there is a P ∈ N ( G ) such that P ⊆ U and P ∩ G ◦ ⊆ QC . Since C contains no nontrivial subgroups, we conclude that P ∩ G ◦ ⊆ Q (because pr C ( P ∩ G ◦ ) is a subgroup of C ). Since G/P is a Lie group and
G/G ◦ is compact, G/P G ◦ is finite. Thus P G ◦ is open in G , and we may as well assume that G = P G ◦ . Thegroup γ ( L ) centralizes Q and normalizes P ✂ G . Therefore it normalizes the compact group N = P Q ⊆ G . We put α : L ✲ Aut( N ) , α ( ℓ )( n ) = γ ( ℓ ) nγ ( ℓ ) − Then the semidirect product N ⋊ α L has a continuous homomorphism ϕ : N ⋊ α L ✲ G, ϕ ( n, ℓ ) = nγ ( ℓ ) . Its image contains P , Q and γ ( L ). Therefore it maps onto P Qγ ( L ) = P G ◦ = G . Since L and N are σ -compact, the group N ⋊ α L is σ -compact. By the Open Mapping Theoremfor Locally Compact Groups (see e.g. [20] p. 669), ϕ is open. We claim that the kernel isdiscrete. Let W = γ − ( C ). Then N × W is an identity neighborhood of N ⋊ α L . Suppose that( n, w ) ∈ ( N × W ) ∩ ker ϕ . Then n = γ ( w ) − = c ∈ C and n = qp for some p ∈ P and q ∈ Q .Thus p = q − c ∈ QC ⊆ G ◦ . It follows that p ∈ P ∩ G ◦ ⊆ Q . Thus we may assume that p = 1,and this implies n = q = c = 1. Thus ϕ has a discrete kernel. Next we note that N ◦ ⊆ G ◦ , andthus N ◦ ⊆ Q . Therefore L centralizes N ◦ . By Lemma 4.3, we have an isomorphism N × L ✲ N ⋊ α L which is the identity on N × { } . Thus we have proved Claim 2. Claim 3. The result holds for arbitrary locally compact groups G . In such a group G , there isan open subgroup H ⊆ G such that H/H ◦ is compact. By Claim 2 we find a simply connectedLie group L , a compact subgroup N and an open homomorphism N × L ✲ H ⊂ ✲ G. ✷ We now extend our results from Section 3 to locally compact groups.
By way of comparison we remind the reader that a space X is called locally contractible if it satisfies the following condition. (LC*) For every point x ∈ X and every neighborhood V of x there exists a neighborhood U ⊆ V of x which is contractible in V .A locally contractible space is locally arcwise connected and piecewise contractible, that is,13LC*) = ⇒ (PC).A neighborhood retract of a locally contractible space is locally contractible, see [17] Theorem4–42 or [24] I.9. It follows that a product of two spaces is locally contractible if and only ifthe two factors are locally contractible. We also note that being locally contractible is a localproperty of a space.In view of Theorem 4.4 the following lemma is the main step in this extension. Let L be a Lie group and let N be a compact group.Suppose that K is a closed subgroup of the locally compact group G = N × L . If X = G/K islocally contractible, then X is a manifold. If N acts faithfully on X , then N is a compact Liegroup and hence G is a Lie group. Proof.
Since N is compact and normal in G , the group N K ⊆ G is closed, see [16] II 4.4.Because N is a direct factor in G , the group N K splits as
N K = N × H , and H ⊆ L is aclosed Lie subgroup. The natural map N × H ✲ N × L ( N × L ) / ( N × H ) ❄ is a locally trivial principal bundle because this is true for the map L ✲ L/H , see [35],Theorem 3.58. It follows that the associated bundle( N × H ) /K ✲ ( N × L ) /K ( N × L ) / ( N × H ) ❄ is also locally trivial. Since ( N × L ) /K is locally contractible, the same is true for the fiber( N × H ) /K by the remarks in 5.1. Now the compact group N acts transitively on the fiber F = ( N × H ) /K = N K/K ∼ = N/N ∩ K. By Theorem 3.5, the homogeneous space F is a closed manifold. Since the base space B =( N × L ) / ( N × H ) ∼ = L/H is also a manifold and since X = ( N × L ) /K is locally homeomorphicto B × F , we have that X is a manifold. If N acts faithfully on G/K , then N acts faithfullyon F , because a subgroup P ⊆ N ∩ K which is normal in N is also normal in N × L . Hence N is a compact Lie group in this case. ✷ Let G be a locally compact group and let K ⊆ G be a closed subgroup. If X = G/K is locally contractible, then X is a manifold. Proof.
Let ϕ : N × L ✲ G be as in Theorem 4.4. Then H = ϕ ( N × L ) is an opensubgroup of G . Therefore H has an open orbit Y = HK/K ⊆ X . By Lemma 5.2, this openset Y is a manifold. It follows from the homogeneity of X that X itself is a manifold. ✷ Let G be a locally compact group and let K ⊆ G be a closed subgroup. Assumethat X = G/K is locally contractible and that X is connected or that G/G ◦ is compact. If G acts faithfully on X , then G is a Lie group. Proof.
Assume first that X is connected. Then we argue similarly as in the proof ofCorollary 5.3 and we consider the open subgroup H = ϕ ( N × L ). The H -orbit of the coset gK ∈ G/K is the open set
HgK/K . Because X is connected, this implies that H acts transitivelyon X . In particular, N ⊆ H acts faithfully on H/H ∩ K ∼ = G/K . By Lemma 5.2, H is a Liegroup. Since H ⊆ G is open, G is also a Lie group. If G/G ◦ is compact, then G is a Lie groupby [28] Ch. 6.3, Corollary on p. 243. ✷ Again, we restate this result in terms of transformation groups.
Let G be a locally compact group, with G/G ◦ compact. If X is a locallycontractible, locally compact space and if G × X ✲ X is a transitive continuous faithfulaction, then G is a Lie group and X is a manifold. Proof.
Let x ∈ X be a point and let K ⊆ G denote the stabilizer of x . Our assumptions im-ply that G is σ -compact, hence the natural continuous map G/K ✲ X is a homeomorphism,see for example [33] 10.10. ✷ In his influential 1974 paper [34] Szenthe stated the following result on locally compact groups:
Theorem 4 [34]. Let a σ -compact group G with compact G/G ◦ be an effective and transitivetopological transformation group of a locally compact and locally contractible space X . Then G is a Lie group and X is homeomorphic to a coset space of G . Szenthe’s statement provided a result which was needed and applied in various areas, notablyin geometry, see for example [1, 4, 5, 31, 14, 15, 26]. The proof of Theorem 4 was based, amongother things, on the following statement on compact groups.
Lemma 6 [34]. Let G be a compact group, H ⊆ G a closed subgroup, χ : G ✲ G/H thecanonical projection, A ⊆ G a closed invariant subgroup and A ′ = χ ( A ) . If A ′ is contractibleover G/H , then A ⊆ H . However, in 2011, Sergey Antonyan [2] found the following simple counterexample to Szenthe’sLemma 6:
Example.
Let G = S ⊆ C ∗ , let H = { } and and put A = {± } . Then A ′ = A is contractiblein G , but is not contained in H .It was also noted the mid-nineties by Salzmann and his school [31] that Szenthe’s method ofapproximating a locally compact group G by Lie groups forces G to be metric, that is, firstcountable, see [6]. Thus even if a substitute method for Szenthe’s Lemma 6 for compact groups15ould be obtained by some correct argument, the ensuing version of Theorem 4 could only bevalid for first countable locally compact groups. References [1] S. M. Ageev and D. Repovsh, On the extension of actions of groups (Russian), Mat. Sb. (2010), no. 2, 3–28; translation in Sb. Math. (2010), no. 1-2, 159–182. MR2656322(2011h:54050)[2] S. A. Antonyan, Characterizing maximal compact subgroups, Arch. Math. (Basel) (2012), no. 6, 555–560. MR2935661[3] S. A. Antonyan and T. Dobrowolski, Locally contractible coset spaces, to appear in: ForumMath.[4] V. N. Berestovski˘ı, Similarly homogeneous locally complete spaces with an intrinsic metric(Russian), Izv. Vyssh. Uchebn. Zaved. Mat. , no. 11, 3–22; translation in RussianMath. (Iz. VUZ) (2004), no. 11, 1–19 (2005). MR2179443 (2006i:53055)[5] V. N. Berestovski˘ı, D. M. Halverson and D. Repovˇs, Locally G -homogeneous Busemann G -spaces, Differential Geom. Appl. (2011), no. 3, 299–318. MR2795840 (2012g:53153)[6] H. Bickel, Lie-projective groups, J. Lie Theory (1995), no. 1, 15–24. MR1362007(96k:22009)[7] G. E. Bredon, Introduction to compact transformation groups , Academic Press, New York,1972. MR0413144 (54
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60 (102) (1963), 63–88. MR0147575 (26 (1964), 57–82.[33] M. Stroppel, Locally compact groups , EMS Textbooks in Mathematics, European Mathe-matical Society (EMS), Z¨urich, 2006. MR2226087 (2007d:22001)[34] J. Szenthe, On the topological characterization of transitive Lie group actions, Acta Sci.Math. (Szeged) (1974), 323–344. MR0360921 (50 Foundations of differentiable manifolds and Lie groups , corrected reprintof the 1971 edition, Graduate Texts in Mathematics, 94, Springer, New York, 1983.MR0722297 (84k:58001)[36] H. Yamabe, On the conjecture of Iwasawa and Gleason, Ann. of Math. (2) (1953),48–54. MR0054613 (14,948e)Karl H. HofmannTechnische Universit¨at DarmstadtSchlossgartenstraße 764289 Darmstadt, Germany [email protected] Linus KramerMathematisches InstitutEinsteinstr. 6248149 M¨unster, Germany [email protected] rratum to:Transitive actions of locally compact groupson locally contractible spaces Karl H. Hofmann and Linus KramerAfter the article
Transitive actions of locally compact groups on locally contractible spaces went to online publication, we noticed that the proof of Lemma 3.4 is incorrect. The problem isthat a lift of a homotopically constant map in a fibration need not be homotopically constant.This issue can be resolved as follows. The numbering and the references are as in our originalarticle. We have added the references [37] and [38].We first prove a preparatory Lemma about coverings of piecewise contractible spaces. Wecall a map E p ✲ B a covering map if every point b ∈ B has a neighborhood V which is evenlycovered, i.e. p − ( V ) p ✲ V is isomorphic to the product map V × F ✲ V , for some discretespace F . We do not impose (local) connectivity assumptions on B or E . Lemma 3.1 Suppose that E p ✲ B is a covering map. If B is piecewise contractible, then E is piecewise contractible.Proof. (See [38], Lemma 10.77.) Let e ∈ E be a point, and let U be an open neighborhoodof p ( e ) which is evenly covered. Replacing U by a smaller neighborhood of p ( e ) if necessary, wemay assume that U is contractible in B by a homotopy h : U × [0 , ✲ B . Let s : U ✲ E be a cross section of p over U such that s ( U ) is a neighborhood of e . A covering map isautomatically a fibration [37, Theorem I.7.12], hence there exists a lift ˜ h : U × [0 , ✲ E of h with ˜ h = s . Then ˜ h maps U into the discrete fiber F = p − ( h ( p ( e ))). The preimage V of˜ h ( e ) is therefore open in U . Thus s ( V ) is an open neighborhood of e which can be contractedin E . ✷ Lemma 3.4
Let
G, K, X, M be as in Lemma 3.3. If G acts faithfully on X , then M is acompact Lie group.Proof. (See [38], Proof of Lemma 10.78.) We put L = M ∩ K and we identify X with thequotient space M/L . It follows from Theorem 2.7 that there is a compact connected semisimpleLie group S , a compact connected finite dimensional abelian group A , and a surjective homo-morphism q : A × S ✲ M with a totally disconnected kernel E . We may assume that q maps A isomorphically onto Cen( M ) ◦ . Then E is finite and q is a covering homomorphism. We notethat L intersects q ( A ) = Cen( M ) ◦ trivially, because the action is faithful. Hence L injects intothe compact semisimple Lie group M/q ( A ). In particular, L is a compact Lie group. Since q is a covering homomorphism, the group H = q − ( L ) is also a compact Lie group. Then thecompact connected group ( { } × S ) H ◦ = T × S ⊆ A × S is also a Lie group by Lemma 2.3, and19herefore T is a finite dimensional torus. It follows from [19, Theorem 8.78(ii)] that A splits as A ∼ = T × B , for some compact abelian group B . Now we have ( A × S ) /H ◦ ∼ = B × (( T × S ) /H ◦ ).By Lemma 3.1 , the space ( A × S ) /H ◦ is piecewise contractible, and by Lemma 3.1, B is piece-wise contractible. In particular, the path component of the identity in B is open in B . Since B is connected, it is therefore path connected. Since B has finite dimension, it is metrizable [21,Theorem 8.49] and therefore a finite dimensional torus [21, Theorem 8.46(iii)]. Thus A itself isa finite dimensional torus, and A × S is a compact Lie group. Then M = q ( A × S ) is also aLie group. ✷ References [37]
G. W. Whitehead , Elements of homotopy theory, Graduate Texts in Mathematics .Springer. New York. 1978.[38] K. H. Hofmann and S. A. Morris , The structure of compact groups, 3rd ed., de GruyterStud. Math. . Walter de Gruyter. Berlin 2013.Karl H. HofmannTechnische Universit¨at DarmstadtSchlossgartenstraße 764289 Darmstadt, Germany [email protected] Linus KramerMathematisches InstitutEinsteinstr. 6248149 M¨unster, Germany [email protected]@uni-muenster.de