On homeomorphism groups of non-compact surfaces, endowed with the Whitney topology

This paper is devoted to studying the topological structure of (the identity component H 0 (M ) of) the homeomorphism group H(M ) of a non-compact connected surface M . By a surface we understand a σ-compact 2-manifold M possibly with boundary ∂M . By [13], each surface M admits a combinatorial triangulation, unique up to PL-homeomorphisms. If necessary, we fix a triangulation of M and regard M as a PL 2-manifold. A subpolyhedron of M means a subpolyhedron with respect to a PL-structure on M . For a surface M let H(M ) denote the homeomorphism group of M endowed with the Whitney topology. This topology is generated by the base consisting of the sets Given a subset K ⊂ M , consider the subgroup Let H 0 (M ; K) be the identity connected component of H(M ; K). This is the largest connected subset that contains the neutral element id M of the group H(M ; K). By [4, Proposition 3.3] H 0 (M ; K) lies in the subgroup H c (M ; K) of H(M ; K) that consists of all homeomorphisms f ∈ H(M ; K) having compact support supp We write H(M ), H 0 (M ) and H c (M ) instead of H(M ; ∅), H 0 (M ; ∅), and H c (M ; ∅). The local topological structure of the groups H 0 (M ; K) and H c (M ; K) was studied in [4] for the case when K is a subpolyhedron in a surface M . It was shown in [4] that the topological group Another result of [4] says that the topological group H c (M ; K) is locally contractible and hence the connected component H 0 (M ; K) is an open subgroup of H c (M ; K). Consequently, H c (M ; K) is homeomorphic to the product H 0 (M ; K) × M c (M ; K) of the connected group H 0 (M ; K) and the discrete quotient group M c (M ; K) = H c (M ; K)/H 0 (M ; K), which can be called the mapping class group of the pair (M, K). Therefore, the topological type of the group H c (M ; K) is completely determined by that of H 0 (M ; K) and the cardinality of M c (M ; K). The topological structure of the group H 0 (M ; K) is well-understood in case of a compact connected surface M . The following classification result belongs to M. Hamstrom [8] (cf. [18]). Theorem 1 (Hamstrom). For a subpolyhedron K M in a compact connected surface M the group Here D denotes the disk, A = D#D the annulus, P the projective plane, M = P#D the Möbius band, K = P#P the Klein bottle, and T the torus. As expected, S n denotes the n-dimensional sphere. For two surfaces M, N by M #N we denote their connected sum. The connected sum M #D is homeomorphic to M with removed open disk. The classification Theorem 1 is completed by the following theorem that will be proved in Section 3. Theorem 2. For a subpolyhedron K M in a non-compact connected surface M the group For non-compact graphs a counterpart of Theorem 2 was proved in [3]. Next, we calculate the cardinality of the mapping class group M c (M ) = H c (M )/H 0 (M ) of a connected surface M . For compact connected surfaces the following classification is known, see [5], [7, Ch.2], [9], [10], [11], [15], [16, §4.4]. Theorem 3. For a compact connected surface M the mapping class group M c (M ) is isomorphic to Here D 2n = a, b | a n = b 2 = baba = 1 is the dihedral group (of isometries of the regular n-gon) having 2n elements. It should be mentioned that the group D 2 is isomorphic to ( ∼ =) the cyclic group We complete Theorem 3 by the following theorem that will be proved in Section 4. Theorem 4. For a non-compact connected surface M , the following conditions are equivalent: (3) M is homeomorphic to X \ K, where X = A, D or M, and K is a non-empty compact subset of a boundary circle of X. To distinguish the surfaces appearing in Theorems 1, 3, 4, let us introduce the following terminology: Definition 1. A surface M is called exceptional if it is homeomorphic to one of the surfaces: S 2 , T, P, K, M#D, D#D#D or X \ K, where X = A, D or M and K is a (possibly empty) compact subset of a boundary circle of X. Unifying Theorems 1-4, we get the following table describing all possible topological types of the homeomorphism groups H 0 (M ) and H c (M ) of a connected surface M . In this table cardinal numbers (identified with the sets of smaller ordinals) are endowed with the discrete topology. It is interesting to compare the classification Theorem 2 with the following result on the compact-open topology due to the last author [18]. (1) (2) l 2 in all other cases. Recognizing topological groups homeomorphic to R ∞ × l 2 In this section we recall a criterion for recognizing topological groups homeomorphic to R ∞ × l 2 , which is proved in [2] and is based upon the topological characterization of the space R ∞ × l 2 given in [1]. First we recall some necessary definitions. A subgroup H of a topological group G is called locally topologically complemented (LTC) in G if H is closed in G and the quotient map q : G → G/H = {xH : x ∈ G} is a locally trivial bundle, which happens if and only if q has a local section at some point of G/H. Here, a local section of a map q : X → Y at a point y ∈ Y means a continuous map s : U → X defined on a neighborhood U of y in Y such that q • s = id U . (As usual, id U denotes the inclusion map U ⊂ Y .) We understand that the distinguished point of a group G is its neutral element e, and for a subgroup H the distinguished element of the quotient space G/H is the coset ē = eH. Lemma 1. Suppose G is a topological group and K ⊂ H are closed subgroups of G. (1) If G is metrizable, then so is the quotient space G/H. ( (2) By the assumption, the projection G → G/H has a local section σ : (W, e) → (G, e) at e ∈ G/H and the projection H → H/K also has a local section τ : U → H at e ∈ H/K. Consider the projection π : G/K → G/H. The map σ determines the map σ 0 : π -1 (W ) → H/K, σ 0 (x) = σ(π(x)) -1 x. Since σ 0 (e) = e ∈ U , there exists an open neighborhood V of e in π -1 (W ) such that σ 0 (V ) ⊂ U . The required local section s : V → G for the projection G → G/K is defined by s(x) = σ(π(x))τ (σ 0 (x)). (3) The projection G → G/H has a local section σ : W → G at any point x 0 ∈ G/H. The associated trivialization φ : W × H/K ≈ π -1 (W ) is defined by φ(x, y) = σ(x)y. Remark 1. We can also show the following statements. Suppose H is a closed subgroup of G and K is an open normal subgroup of H. (1) The natural map ξ : G/K → G/H, ξ(gK) = gH, is a covering map with fiber H/K. (2) H is locally topologically complemented in G if and only if so is K in G. Following [2], we say that a topological group G carries the strong topology with respect to a tower of subgroups 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. A point x of X is called a (strong) Z-point if the singleton {x} is a (strong) Z-set in X. Every point in an infinite-dimensional Hilbert manifold is a strong Z-point. The following theorem proved in [2] will be our main instrument in the proof of Theorem 2. Theorem 6. A non-metrizable topological group G is homeomorphic to the space R ∞ ×l 2 if G carries the strong topology with respect to a tower of subgroups (G n ) n∈ω such that each group G n is homeomorphic to l 2 , is locally topologically complemented in G n+1 , and each Z-point of the quotient space G n+1 /G n is a strong Z-point. Suppose M is a non-compact connected surface and K M is a subpolyhedron of M (with respect to some PL-structure τ of M ). The boundary and interior of M as a manifold are denoted by ∂M and Int M , while the topological closure, interior and frontier of a subset A in M are denoted by the symbols cl M A, int M A and bd M A respectively. Case (1) follows from the following lemma. Proof. There exists a compact connected PL 2-submanifold N of (M, τ ) such that cl M (M \K) ⊂ N . By the Hamstrom's Theorem 1, H 0 (N ; N ∩ K) ≈ l 2 . In fact, N ∩ K contains the frontier bd M N , which is a 1-manifold and is nonempty since M is connected. By [18] H(N ; N ∩K) is an l 2 -manifold (in particular, it is locally connected). The conclusion now follows from (H(M ; K), id M ) ≈ (H(N ; N ∩ K), id N ). Below we assume that cl M (M \ K) is non-compact. Then the surface M can be represented as the countable union M = n∈ω M n of compact subpolyhedra M n , n ∈ ω, of (M, τ ) such that M n ⊂ int M M n+1 and int M M n ⊂ K for each n ∈ ω. Consider the subpolyhedra Mn = M \ int M M n and K n = K ∪ Mn (n ∈ ω) of (M, τ ). Then we obtain the subgroups H = H c (M ; K) and Our main concern is the identity connected components G = H 0 (M ; K) and G n = H 0 (M ; K n ) (n ∈ ω). Note that G is not metrizable and it coincides with the identity connected component of the group H. In Lemmas 3 -5 below we show that the tower of subgroups G n , n ∈ ω, of the group G satisfies the conditions in Theorem 6. This implies Case (2). Proof. (1) First note that G = H 0 (M ; K) is path-connected since the group H c (M ; K) is locally pathconnected by [4,Theorem 6.5] and G is the identity connected component of H c (M ; K). Hence any h ∈ G can be joined to id M by an arc A in G. By [4, Proposition 3.3] the compact subset A lies in H(M ; Mn ) for some n ∈ ω. Since (2) The assertion follows from Lemma 2 since K n M and M \ K n is included in the compact subset M n . To check the LTC condition and the Z-point property in Theorem 6 for the tower (G n ) n∈ω we need a result on embedding spaces. For closed subsets ) is endowed with the compact open topology. The following basic fact is due to Yagasaki [17], and in its present form can be found in [4,Theorem 6.7]. One should note that the assertion was verified in [12] in the important case that K = ∅ and L is either a proper arc, an orientation-preserving circle or a compact 2-submanifold of M . Theorem 7 (Yagasaki). Suppose L ⊂ N are two subpolyhedra of a surface M with compact closure cl M (N \ K). (1) The restriction map The subgroup H m is LTC in H n and the quotient space H n /H m is an l 2 -manifold for any m ≤ n in ω. (2) The subgroup G m is LTC in G n and the quotient space G n /G m is an l 2 -manifold (in particular, every point of G n /G m is a strong Z-point) for any m ≤ n in ω. Proof. (1) The group H n = H(M ; K n ) acts continuously on the space E * Kn (K m , M ) by the left composition. Under this action the subgroup H m is the isotropy group of id Km and the restriction map ), R : h → h| Km , coincides with the orbit map at id Km . Hence we have the factorization where the map r is the natural projection and the induced map φ is defined by φ(hH) = h| Km . Note that the map φ is a homeomorphism since φ is bijective and both maps r and R are open continuous surjections. Therefore, by Theorem 7 the map r has a continuous section and H n /H m is an l 2 -manifold. ( It remains to check the strong topology condition for the tower (G n ) n∈ω . This would be derived from Theorem 7 and general arguments on transformation groups with strong topology in [4, Section 5]. However, for the convenience of the reader, here we include its direct self-contained proof and avoid exceeding generality on related subjects. Let us recall some notation. For covers U , V of a space M and a subset For two maps f, g : X → M we write (f, g) ≺ U if for each point x ∈ X the doubleton {f (x), g(x)} lies in some subset U ∈ U . By Proof. We may assume that each U n is symmetric (i.e., U n = U -1 n ). It suffices to show that the set is a neighborhood of id M in H c (M ; K). By Theorem 7, for every n ∈ ω the restriction map R n : G n → E * Kn (K n-1 , M ), R n : h → h| K n-1 has a local section s n : (V n , id K n-1 ) → (G n , id M ) at id K n-1 . Inductively we can find covers U n , V n ∈ cov(M ), n ∈ ω, satisfying the next conditions for each n ∈ ω: (1) (i) St(U n ) ≺ V n-1 (where V -1 = {M }); (ii) If h ∈ H(M ; K n ) and (h, id M ) ≺ U n , then h ∈ U n ; (