Detecting topological groups which are (locally) homeomorphic to LF-spaces

The problem of recognizing the topological structure of topological groups traces its history back to the fifth problem of Hilbert which asks if Lie groups can be characterized as topological groups whose underlying topological spaces are manifolds. This problem was resolved by combined efforts of Gleason [17], Montgomery and Zippin [24], Hoffman [19], and Iwasawa [16] who proved the following of Fréchet (i.e., locally convex linear completely metrizable) spaces in the category of locally convex spaces. More precisely, lc-lim -→ X n is the union X = n∈ω X n endowed with the strongest topology that turns X into a locally convex space and makes the identity inclusions X n → X, n ∈ ω, continuous. The simplest non-trivial example of an LF-space is R ∞ , the direct limit of the tower where each space R n is identified with the hyperplane R n × {0} in R n+1 . The space R ∞ can be identified with the direct sum n∈ω R of one-dimensional Banach spaces in the category of locally convex spaces. The topological classification of LF-spaces was obtained by Mankiewicz [21] who proved that each LF-space is homeomorphic to the direct sum n∈ω l 2 (κ i ) of Hilbert spaces for some sequence of cardinals (κ i ) i∈ω . Here l 2 (κ) denotes the Hilbert space with an orthonormal base of cardinality κ. A more precise version of Mankiewicz's classification says that the following spaces • l 2 (κ) × R ∞ for some κ ≥ ω, and • n∈ω l 2 (κ i ) for a strictly increasing sequence of infinite cardinals (κ i ) i∈ω are pairwise non-homeomorphic and represent all possible topological types of LF-spaces. In particular, each infinite-dimensional separable LF-space is homeomorphic to one of the following spaces: l 2 , R ∞ or l 2 × R ∞ . The topological characterizations of the LF-spaces l 2 and R ∞ were given by Toruńczyk [28], [29] and Sakai [25], respectively. Other LF-spaces were recently characterized by Banakh and Repovš [5]. The description of the topology of the direct sum n∈ω X n of locally convex spaces given in [26, II. §6.1] implies that this topology coincides with the topology of the small box-product ⊡ n∈ω X n . The construction of the small box-product ⊡ n∈ω X n of pointed topological spaces is purely topological and is defined as follows. By a pointed space X we understand a space with a distinguished point, which will be denoted by * X . Each group G is a pointed space whose distinguished point * G is the neutral element of G. For a subgroup H ⊂ G the quotient space G/H = {xH : x ∈ G} is a pointed space with the distinguished point The small box-product of a sequence (X n ) n∈ω of pointed topological spaces is the subspace of the box-product n∈ω X n . The latter space is the product n∈ω X n endowed with the topology generated by the products n∈ω U n of open subsets U n ⊂ X n , n ∈ ω. Now let us return to the problem of recognizing topological groups that are (locally) homeomorphic to LFspaces. We say that a topological group G carries the strong topology with respect to a tower of subgroups the group G. The nature of this property will be discussed in Section 3. A closed subgroup H of a topological group G is defined to be (locally) topologically complemented in G if the quotient map π : G → G/H, π : x → xH, is a (locally) trivial bundle. This happens if and only if π has a section s : G/H → G, which is continuous on (some non-empty open subset of) the quotient space G/H. It follows that for a (locally) topologically complemented subgroup H of G the group G is (locally) homeomorphic to the product H × (G/H). In Proposition 4.2 we shall prove that a locally topologically complemented subgroup H of an ANR-group G is topologically complemented in G if the quotient space G/H is contractible or both groups G and H are contractible. By an ANR-group we understand a topological group whose underlying topological space is an ANR. Therefore, each ANR-group is metrizable. A tower of groups (G n ) n∈ω is called (locally) The topology of a topological group carrying the strong topology with respect to a (locally) topologically complemented tower of subgroups is closely related to small box-products: Theorem 1.4. A topological group G carrying the strong topology with respect to a (locally) topologically complemented tower of subgroups (G n ) n∈ω is (locally) homeomorphic to the small box-product Theorem 1.4 will be proved in Section 5. This theorem motivates the problem of studying the topological structure of small box-products and recognizing small box-products that are (locally) homeomorphic to LFspaces. A corresponding criterion was proved in [5]. It involves the notion of a strong Z-point. Let us recall that a closed subset A of a topological space X is called a (strong) Z-set in X if for any open cover U of X there is a continuous map f : X → X such that f is U-near to the identity id X : X → X and (the closure f (X) of) the set f (X) does not intersect A. It is clear that each strong Z-set is a Z-set. The converse is not true, see [12]. However each Z-set in a Hilbert manifold is a strong Z-set, see [11], [29]. A point x of a topological space X will be called a (strong) Z-point if the singleton {x} is a (strong) Z-set in X. A pointed topological space X is called lz-pointed if the distinguished point * X is not isolated in X and either X is locally compact or * X is a strong Z-point in X. We shall use the following criterion proved in [5]: Theorem 1.5 (Banakh-Repovš). The small box-product ⊡ n∈ω X n of pointed topological spaces X n , n ∈ ω, is homeomorphic to (an open subspace of ) an LF-space if for every n ∈ ω the finite product i≤n X i is homeomorphic to (an open subset of ) a Hilbert space and the space X n is lz-pointed for infinitely many numbers n ∈ ω. We shall say that a topological space has the Z-point property if each Z-point in X is a strong Z-point. Since each Z-set in a Hilbert manifold is a strong Z-set, Theorem 1.2 implies that each Polish ANR-group has the Z-point property. In Proposition 6.3 we shall prove that for a locally topologically complemented subgroup H of an ANR-group G the quotient space G/H is lz-pointed if and only if it is not discrete and has the Z-point property. Combining this fact with Theorems 1.4, 1.5 and Proposition 4.2, we obtain the following criterion, which is the main result of this paper. This criterion has been applied in [1] and [6] for recognizing the topology of some homeomorphism and diffeomorphism groups. Hilbert space, G n is locally topologically complemented in G n+1 , and for infinitely many numbers n ∈ ω the quotient space G n+1 /G n is not discrete and has the Z-point property. Because of the lack of an Open Embedding Theorem for LF-manifolds, we distinguish between LF-manifolds and open subspaces of LF-spaces. This is why we have two separate statements (1) and (2) in Theorem 1.6. It should be mentioned that the topological structure of open subspaces of LF-spaces is quite well understood, which cannot be said about LF-manifolds, see [22], [23]. In light of Theorem 1.6 it is natural to ask if the quotient spaces G n+1 /G n always have the Z-point property. Problem 1.7. Let G be a Polish ANR-group and H be a (locally) topologically complemented subgroup in G. The answer to this problem is (trivially) affirmative if G/H is a Hilbert manifold. This is why Theorem 1.6 implies the following criterion for recognition of topological groups which are locally homeomorphic to LF-spaces. Theorem 1.10 (Banakh-Repovš). If G is a Polish ANR-group and H is a balanced closed ANR-subgroup in G, then the quotient space G/H is a Hilbert manifold and hence has the Z-point property. Combining this theorem with Theorems 1.2, 1.6 and Proposition 4.2, we obtain the following criterion. Theorem 1.11. A topological group G carrying the strong topology with respect to a (locally) topologically complemented tower of Polish ANR-groups Next, we formulate another condition on a subgroup H of a topological group G which implies that the quotient space G/H has the Z-point property. Note that the quotient space G/H is a G-space with the natural left action of the group G. Let us recall that a G-space is a topological space X endowed with a continuous action α : G × X → X, α : (g, x) → gx, of a topological group G. We say that the action of G on X is locally bounded at a point x 0 ∈ X if there is a neighborhood U ⊂ G of the neutral element * G of G such that for every neighborhood V ⊂ X of x 0 there is a compact subset K ⊂ X which meets each shift xV , x ∈ U . In the opposite case, the action is called locally unbounded at x 0 . The action of G on X is locally unbounded if this action is locally unbounded at each point x 0 ∈ X. If the action of G is locally unbounded at some point x 0 ∈ X, then X is not locally compact at x 0 . In Proposition 6.4 we shall show that for a closed subgroup H of a locally path-connected topological group G, each point of the quotient space G/H is a strong Z-point if the space G/H is a separable ANR and the action of G on G/H is locally unbounded. Combining this fact with Theorems 1.2 and 1.6, we get the following criterion: Corollary 1.12. A topological group G carrying the strong topology with respect to a (locally) topologically complemented tower of Polish ANR-groups (G n ) n∈ω is (locally) homeomorphic to an open subset of the LF-space l 2 × R ∞ if for infinitely many numbers n ∈ ω the action of the group G n+1 on G n+1 /G n is locally unbounded. It turns out that the structure of topological groups G carrying the strong topology with respect to a tower of subgroups (G n ) n∈ω can be described in terms of uniform direct limits of towers of uniform spaces. Therefore, in this section we recall the necessary information on this topics. For basic information on uniform spaces we refer the reader to Chapter 8 of Engelking's monograph [15]. All topological spaces considered in this paper are completely regular and all maps are continuous. For a uniform space X we denote its uniformity by U X . Since uniform spaces are completely regular, the intersection ∩U X coincides with the diagonal of X × X. A uniform space X is called metrizable if its uniformity is generated by some metric. Elements of the uniformity U X are called entourages. For an entourage U ∈ U X , a point x ∈ X and a subset A ⊂ X by B(x, U ) = {y ∈ X : (x, y) ∈ U } we denote the U -ball centered at x and by By a tower of uniform spaces we shall understand an increasing sequence of uniform spaces (so the uniformity of each space X n coincides with the uniformity inherited from the uniform space X n+1 ). For a tower of uniform spaces X n is the union X = n∈ω X n endowed with the largest uniformity making the identity inclusions X n → X, n ∈ ω, uniformly continuous. The topology and the uniformity of the uniform direct limit u-lim -→ X n were described in [2]. If each space X n of the tower is locally compact then the topology of u-lim -→ X n coincides with the topology of the topological direct limit t-lim -→ X n of the tower (X n ) n∈ω . The topological direct limit t-lim -→ X n of a tower (X n ) n∈ω of topological spaces is the union n∈ω X n endowed with the largest topology turning the identity inclusions X n → X, n ∈ ω, into continuous maps. If (X i ) n∈ω is a sequence of pointed uniform spaces, then each finite (box-)product carries the product uniformity. Therefore, (⊡ i≤n X i ) n∈ω turns into a tower of uniform spaces whose union n∈ω ⊡ i≤n X i coincides with the small box-product ⊡ n∈ω X i . The following lemma was proved in [2, 5.5]. Lemma 2.1. For a sequence (X i ) i∈ω of pointed uniform spaces the identity map Next, we recall the definition of a (locally) complemented subset of a uniform space, introduced in [5]. for any entourage U ∈ U X there is a neighborhood The following important fact was proved in [5]. Lemma 2.3. Let (Z n ) n∈ω be a sequence of pointed topological spaces and (X n ) n∈ω be a tower of uniform spaces. (1) If each set X n is Z n -complemented in X n+1 , then the uniform direct limit u-lim -→ X n is homeomorphic to the small box-product In this section we shall study the structure of topological groups G that carry the strong topology with respect to a tower of subgroups (G n ) n∈ω . It turns out that this happens if and only if G is the direct limit of this tower in the categories of topological groups or uniform spaces. Let us recall that each topological group G carries four natural uniformities: • the left uniformity U L generated by the entourages • the two-sided uniformity U LR generated by the entourages U LR = {(x, y) ∈ G 2 : x ∈ yU ∩ U y}, and • the Roelcke uniformity U RL generated by the entourages where U = U -1 runs over open symmetric neighborhoods of the neutral element e of G. The group G endowed with one of the uniformities U L , U R , U LR , U RL is denoted by G L , G R , G LR , G RL , respectively. These four uniformities on G coincide if and only if the group G is a SIN-group, which means that G has a neighborhood base at * G consisting of open sets U ⊂ G that are invariant in the sense that Let G be a topological group and (G n ) n∈ω be a tower of closed subgroups of G such that G = n∈ω G n . Endowing the subgroups G n , n ∈ ω, with one of four canonical uniformities, we obtain four uniform direct limits u-lim Besides those direct limits, the group G also carries the topology of the group direct limit g-lim -→ G n of the tower (G n ) n∈ω . This is the strongest topology that turns G = n∈ω G n into a topological group and makes the identity maps G n → G, n ∈ ω, continuous. For these direct limits we get the following diagram in which each arrow indicates that the corresponding identity map is continuous: The following proposition was proved in [4]. Proposition 3.1. For a topological group G and a tower of subgroups (G n ) n∈ω with G = n∈ω G n the following conditions are equivalent: (1) G carries the strong topology with respect to the tower (G n ) n∈ω ; (2) the identity map u-lim (4) the identity map g-lim It should be mentioned that for a tower of metrizable topological groups (G n ) n∈ω the identity map t-lim G n is a homeomorphism if and only if all groups G n are locally compact or there is a number m ∈ ω such that for every n ≥ m the group G n is open in G n+1 , see [4], [8], [30] or [18, 7.1]. If (X n ) n∈ω is a tower of locally convex linear topological spaces, then besides the topology of the group direct limit g-lim -→ X n , the union X = n∈ω X n carries also the topology of the direct limit in the category of (locally convex) linear topological spaces. The corresponding direct limit space is denoted by lc-lim -→ X n (resp. l-lim -→ X n ). This is the union X = n∈ω X n endowed with the strongest topology that turns X into a (locally convex) linear topological space and makes the identity maps X n → X, n ∈ ω, continuous. The following proposition proved in [2] implies that many direct limit topologies on X coincide. Proposition 3.2. For any tower (X n ) n∈ω of locally convex linear topological spaces the identity maps In particular, each LF-space has the topology of the uniform direct limit of a tower of Fréchet spaces. In this section we study (locally) topologically complemented subgroups of topological groups. Let us recall that a closed subgroup H of a topological group is (locally) topologically complemented if the quotient map q : G → G/H is a (locally) trivial bundle. Here G/H = {xH : x ∈ G} is the quotient space of left cosets of H in G. It is a pointed topological space with a distinguished point * G/H = H. For the theory of bundles, we refer the reader to [20]. Proof. Since H is locally topologically complemented in G, the quotient map q : G → G/H has a continuous section s : U → G defined in an open neighborhood U ⊂ G/H of the distinguished point * G/H . If H is topologically complemented in G, then we can take U to be equal to G/H. Replacing s(x) by s(x)s( * G/H ) -1 , we can additionally assume that s( * G/H ) coincides with the neutral element * G of the group G. It follows from the definition of the uniformity U R that the preimage q -1 (U ) is an open uniform neighborhood of H in the uniform space G R . Now we see that the homeomorphism In some cases the local topological complementability implies the topological complementability. Proof. First we show that the quotient space G/H is a (metrizable) ANR. Being metrizable, the group G admits a right invariant metric d generating the topology of G. Then the topology of the quotient space G/H is generated by the metric ρ defined by Hence G/H, being metrizable, is paracompact. Since H is locally topologically complemented in G, the quotient map q : G → G/H is a locally trivial bundle with fiber H. This implies that the space H × (G/H) is locally homeomorphic to G. Since G is an ANR, each point of the quotient space G/H has an ANR-neighborhood, which implies that G/H is an ANR, see Hanner's Theorem 5.1 in [10, Ch.II]. If the quotient space G/H is contractible, then the locally trivial bundle q : G → G/H is trivial according to Corollary 4.10.3 of [20]. Now assume that the spaces G and H are contractible. We claim that the quotient space G/H contractible. Since G is path connected, so is G/H. Since both the total space G and the fiber H of the bundle q : G → G/H are contractible, the exact sequence of the fibration π : G → G/H implies that all homotopy groups π i (G/H) = 0, i ∈ N, of G/H are trivial. Therefore, by Whitehead Theorem (see II.6.1 in [10]), the ANR-space G/H is contractible. In this section we shall prove Theorem 1.4. Assume that a topological group G carries the strong topology with respect to a (locally) topologically complemented tower (G n ) n∈ω of subgroups. By Proposition 3.1, the topology of G coincides with the topology of the uniform direct limit u-lim -→ G R n of the tower (G R n ) n∈ω of the groups G n endowed with their right uniformities. By Lemma 4.1, each set G n is (locally) G n+1 /G n -complemented in G R n+1 . Taking into account that the space G is topologically homogeneous and applying Lemma 2.3, we conclude that G = u-lim -→ G R n is (locally) homeomorphic to the small box-product G 0 × ⊡ n∈ω G n+1 /G n . In this section we shall study the Z-point property in quotient spaces of topological groups. Let us recall that a topological space X has the Z-point property if each Z-point in X is a strong Z-point. In fact, it is more convenient to work not with (strong) Z-point but with an equivalent notion of a (strong) Z ∞ -point. Let κ be a cardinal. A closed subset A of a topological space X is called a (κ×Z ∞ )-set if for each open cover U of X any map f : κ × I ω → X can be approximated by a map f : κ × I ω → X such that f is U-near to f and A does not intersect the closure of the set f (κ × I ω ) in X. We shall refer to (1 × Z ∞ )-sets and (ω × Z ∞ )-sets as Z ∞ -sets and strong Z ∞ -sets, respectively. Such sets were studied in [27], [13, §2.2] and [9, §1.4]. A point x of a topological space X will be called a (strong The following characterization of (strong) Z-sets in (separable) ANR's is well-known, see [27] Proof. Assume that the quotient space G/H is not locally compact and take any compact subset K ⊂ G/H. To prove that K is a Z ∞ -set in G/H, fix an open cover U of G/H and a continuous map f : I ω → G/H from the Hilbert cube. By the local complementability of the subgroup H in G, the quotient map q : G → G/H is a locally trivial bundle. Using this fact and the contractibility of the Hilbert cube I ω , we can find a continuous map g : I ω → G such that q • g = f . By compactness of g(I ω ), there is a neighborhood U ⊂ G of the neutral element * G of G so small that for every u ∈ U the map f u : I ω → G/H defined by f u (x) = q(g(x)u) for x ∈ I ω is U-near to f . Since the quotient map q : G → G/H is open, the set q(U ) is an open neighborhood of * G/H . By the local triviality of q, there is a compact subset K ⊂ G such that q( K) = K. Consider the compact subset C = g(I ω ) of G and the compact subset q(C -1 K) ⊂ G/H, which does not contain the neighborhood q(U ) of * G/H as G/H is not locally compact at * G/H . Consequently, there is an element u ∈ U with q(u) / ∈ q(C -1 K), which implies that u / ∈ C -1 KH and hence Cu ∩ KH = ∅. Then the map f u : I ω → G/H has the required property: it is U-near to f and f u (I ω ) ∩ K = q(g(I ω ) • u) ∩ q( KH) = ∅. Finally, we consider the problem of detecting quotient spaces G/H all whose points are strong Z-points. Let us recall that an action of a topological group G on a topological space X is called locally bounded at a point x 0 ∈ X if there is a neighborhood U ⊂ G of the neutral element * G of G such that for every neighborhood V ⊂ X of x 0 there is a compact subset K ⊂ X that meets each shift xV , x ∈ U . In the opposite case the action is called locally unbounded. . Since G is locally path-connected, there is a neighborhood V 3 ⊂ G of * G such that each point x ∈ V 3 can be connected with * G by a continuous path lying in V 2 . Since the action of G in G/H is locally unbounded, for the neighborhood V 3 there is a neighborhood U 3 ⊂ U 2 of * G/H such that for any compact subset K ⊂ G/H there is a point x ∈ V 3 such that the shift xU 3 does not intersect K. Now we shall construct a map f : ω × I ω → G/H such that f is U-near to f and f (ω × I ω ) ∩ U 3 = ∅. It suffices for every n ∈ ω to approximate the map f n : I ω → G/H, f n : t → f (n, t), by a map fn : I ω → G/H such that fn is U-near to f n and fn (I ω ) ∩ U 3 = ∅. For every n ∈ ω consider the compact subset K n = f ({n} × I ω ) of G/H. By the choice of U 3 there is a point x n ∈ V 3 such that x n U 3 ∩ K n = ∅. Then x -1 n K n ∩ U 3 = ∅. By the choice of the neighborhood V 3 there is a continuous path γ n : [0, 1] → V 2 such that γ n (0) = * G and γ n (1) = x n . Define the map fn : I ω → G/H by the formula fn (t) = γ n (λ(f n (t))) -1 f n (t) for t ∈ I ω . Claim 6.5. The map fn is U-near to f n . n is homeomorphic to (an open subset of ) a Hilbert space; (2) is (locally) homeomorphic to an LF-space if for every n ∈ N the group G n is locally topologically complemented in G n+1 and the quotient space G n+1 /G n is (locally) homeomorphic to a Hilbert space. Considering Theorem 1.8, we can ask another open Problem 1.9. Let G be a Polish ANR-group and H be a (locally) topologically complemented subgroup in G. Is G/H a Hilbert manifold?