Topological (prod^omega ell_2, sum^omega ell_2)-factors of diffeomorphism groups of non-compact manifolds

This article is a continuation of study of topological properties of groups of homeomorphisms and diffeomorphisms of non-compact manifolds with the compact-open (C ∞ -) topology [21,22,23,24]. Suppose G is a transformation group acting on a space M continuously and effectively. Each g ∈ G induces a homeomorphism ĝ of M . When M is non-compact, the group G contains the normal subgroup G c consisting of g ∈ G such that ĝ has a compact support. Let G 0 and (G c ) 0 denote the connected components of the unit element e in G and G c respectively. In this paper we are concerned with the topological type of the pair (G 0 , (G c ) 0 ) in the case where G has a weak topology. Typical examples of the transformation groups G are the group H(M ) of homeomorphisms of a topological manifold (or a locally compact polyhedron) M endowed with the compact-open topology and the group D(M ) of diffeomorphisms of a smooth manifold M endowed with the compact-open C ∞ -topology. In [2] and [3] it is shown that both the pairs (H(M ) 0 , H c (M ) 0 ) for any countable infinite locally finite connected graph M and (D(R) 0 , D c (R) 0 ) for the real line R are homeomorphic to the pair ω ℓ 2 , ω ℓ 2 . Here ω ℓ 2 is the countable product of the separable Hilbert space ℓ 2 and ω ℓ 2 is the countable weak product of ℓ 2 defined by ω ℓ 2 = (x i ) i ∈ ω ℓ 2 | x i = 0 except finitely many i . In this paper, we show that the pairs (H(M ) 0 , H c (M ) 0 ) and (D(M ) 0 , D c (M ) 0 ) for a non-compact 2manifold M are ω ℓ 2 , ω ℓ 2 -manifolds (Theorem 1.2) and determine their topological types from their homotopy types (Corollary 1.1). To establish these results, first we deduce a characterization of ( ω ℓ 2 , ω ℓ 2 )-manifolds under the stability property (Theorem 2.2) from a general criterion [20,Theorem 2.9]. A pair (X, A) is said to be ( ω ℓ 2 , ω ℓ 2 )-stable if (X × ω ℓ 2 , A × ω ℓ 2 ) ∼ = (X, A). Stability properties of homeomorphism groups of topological manifolds and their subgroups have already been studied by many authors (cf. [9,10,12,13,18] etc). In particular, in [22] we have treated the non-compact case in detail. On the other hand, the Moser's theorem for volume forms [16] (cf. [23]) exhibits the ℓ 2 -stability property of diffeomorphism groups. We modify these arguments and show the ( ω ℓ 2 , ω ℓ 2 )-stability property of the pairs (H(M ), H c (M )) and (D(M ), D c (M )) for non-compact (separable metrizable) n-manifolds M . Theorem 1.1. The pair (G, G c ) is ω ℓ 2 , ω ℓ 2 -stable in the following cases: (1) G = D(M ) for a non-compact smooth n-manifold M possibly with boundary (n ≥ 1). (2) G = H(M ) for a non-compact topological n-manifold M possibly with boundary (n ≥ 1). ( and (H(M ) 0 , H c (M ) 0 ). For a subgroup H of a transformation group G on M we use the following notations: component of the unit element e in H, and set Theorem 1.2. Suppose G is one of the following groups: (1) D X (M ) for a non-compact connected smooth 2-manifold M and a compact submanifold X of M . (2) H X (M ) for a non-compact connected 2-manifold M possibly with boundary and a compact subpolyhedron X of M with respect to some triangulation of M . Then the pair Note that the subgroups H = (G c ) * 1 , (G c ) 0 and (G 0 ) c satisfy the conditions in Theorem 1.2. The topological type of any ( ω ℓ 2 , ω ℓ 2 )-manifold (X, A) is classified by the homotopy type of X (Theorem 2.2). Hence, by [21,Theorem 1.1] and [24,Theorem 1.1] we have the conclusion on the global topological type. Consider the next two cases : (II) (M, X) is not the case (I) (in the cases (1) and (2) in Theorem 1.2). Here R n is the Euclidean n-space, S n is the n-sphere and P 2 is the projective plane. This paper is organized as follows. In Section 2 we deduce the characterization of ( ω ℓ 2 , ω ℓ 2 )manifolds based upon the stability property (Theorem 2.2). In Section 3 we obtain the results on the diffeomorphism groups in Theorems 1.1, 1.2 and Corollary 1.1, while Section 4 includes the results on the homeomorphism groups. 2. Characterization of topological ( ω ℓ 2 , ω ℓ 2 )-manifolds In [20] we obtained a general characterization of infinite-dimensional manifold tuples based upon the stability property (cf. [7,15,19], [1,4,5,6], [11], etc.). In this section we deduce a characterization of ( ω ℓ 2 , ω ℓ 2 )-manifolds from this general characterization theorem. 2.1. General characterization of infinite-dimensional manifold pairs under the stability property. We begin with the definition of basic terminology. In this paper spaces are assumed to be separable metrizable and maps are continuous. The symbol ∼ = means a homeomorphism, while ≃ means a homotopy equivalence. A pair of spaces means a pair (X, A) of a topological space X and a subset A of X. We say that two pairs (X, A) and (Y, B) are homeomorphic and write (X, A) ∼ = (Y, B) if there exists a homeomorphism h : X → Y with h(A) = B. For a model space E, an E-manifold means a space X locally homeomorphic to E. More generally, for a model pair (E, E 1 ), by an (E, E 1 )-manifold we mean a pair (X, X 1 ) of spaces such that each point x of X admits an open neighborhood U of x in X and an open subset A closed subset A of a space X is called a Z-set of X if for any open cover U of X there exists a map f : X → X -A which is U -close to id X . A σ Z-set of X means a countable union of Z-sets of X. A subset A of X is said to be homotopy dense (HD) if there exists a homotopy h t : X → X (t ∈ [0, 1]) such that h 0 = id X and h t (X) ⊂ A for t ∈ (0, 1]. Consider the countable product s = k∈N R, which is a topological linear space under the coordinatewise sum and scalar product. Since s is a separable Fréchet space, it follows that s ∼ = ℓ 2 . Suppose s 1 is a linear subspace of s. For I ⊂ N we set c(I) = N \ I and s(I) = k∈I R, and let π I : s → s(I) denote the projection. We set s 1 (I) = π I (s 1 ) ⊂ s(I). Let M ≡ M(s, s 1 ) denote the class of pairs (X, A) which admit a closed embedding h : X → s such that h -1 (s 1 ) = A. Assumption 2.1. We assume that the model pair (s, s 1 ) satisfies the following conditions : ( * 1 ) s 1 is a linear subspace of s and s 1 is a σ Z-set of s 1 itself. ( * 2 ) s 1 is homotopy dense in s. ( * 3 ) There exists a sequence I n (n ≥ 1) of disjoint infinite subsets of N such that for each n ≥ 1 (a) min Under Assumption 2.1 we have the following characterization and homotopy invariance of (s, s 1 )manifolds. This is exactly the case that ℓ = 1 in [20, Theorem 2.9, Corollary 2.10]) Theorem 2.1. A pair (X, A) is an (s, s 1 )-manifold iff (1) X is a separable completely metrizable ANR, (2) (i) (X, A) ∈ M(s, s 1 ), (ii) A is homotopy dense in X, (3) (X, A) is (s, s 1 )-stable. Corollary 2.1. Suppose (X, A) and (Y, B) are (s, s 1 )-manifolds. Then (X, 2.2. Characterization of ( ω ℓ 2 , ω ℓ 2 )-manifolds. Next we deduce a characterization and classification of ω ℓ 2 , ω ℓ 2 -manifolds from Theorem 2.1 and Corollary 2.1. Theorem 2.2. (1) A pair (X, A) is a ω ℓ 2 , ω ℓ 2 -manifold iff it satisfies the following conditions: (i) X is a separable completely metrizable ANR. (2) Suppose (X, A) and (Y, B) are In fact, with replacing the interval [0, 1] by [n, ∞] in the opposite orientation, an absorbing homotopy ψ : , this implies the conditions ( * 1 ) and ( * 2 ) for the pair (s, s 1 ). ( * 3 ) For any infinite subset J of N it is easily seen that the subset Conversely, suppose X is separable completely metrizable and A is F σ in X. Then we can find a closed embedding e : X → s = s 1 ⊂ s ∞ and a map g : X → s such that g -1 (σ) = A, where . Indeed, (i) since e is a closed embedding, so is f , and (ii) since e(X) To treat the groups of homeomorphisms and their subgroups systematically, we formulate our argument to transformation groups. the study of stability property is reduced to seeking for infinite-dimensional factors. In this subsection we give a simple criterion that a G-space admits a product decomposition. Suppose E is a space and G is a topological group which acts continuously on E from the right. We seek a condition that E factors to a product of a subspace F of E and a space B. Consider three maps p : E → B, f : E → F and g : B → G, which induce two maps Lemma 3.1. The maps ϕ and ψ are reciplocal homeomorphisms iff Proof. From the definition of the maps ϕ and ψ, we have the next identities: The condition ( * ) implies that ψϕ(x) = x and ϕψ(b, y) = (b, f (y This means that ψ = ϕ -1 . The converse is obvious. Complement 3.1. In addition, if (a) the maps p, f and g are maps of pairs then the maps ϕ and ψ induce the maps of pairs By Lemma 3.1, if the maps p, f and g satisfy the condition ( * ), then the maps ϕ and ψ are reciplocal homeomorphisms of pairs (and The next lemma is the simplest case of Complement 3.1. Lemma 3.2. Suppose the maps p : E → B, f : E → F and g : B → G satisfy the condition ( * ), so that the map ϕ : Proof. In Complement 3.1 we can take A transformation group on a space M is a topological group G which acts on M continuously and faithfully. Each g ∈ G induces a homeomorphism g of M . For a subset H of G and a subset K of A support function for a space E on M is a function which assigns to each f ∈ E a closed subset supp f of M . When a space E is equipped with a support function on M , for any subspace F of E we obtain the subspace is a normal subgroup of G. Definition 3.1. We say that ( * 1 ) G has a weak topology if for each neighborhood U of e in G there exists a compact subset The element g is denoted by i∈Λ g i . Remark 3.1. Suppose G has the multiplication supported by a discrete family {E i } i∈Λ of compact subsets in M . (1) Since the action of G on M is faithful, each g ∈ G is uniquely determined by g. Thus the element i∈Λ g i is uniquely determined by the defining property. (2) The multiplication map ( , by which the group i∈Λ 0 G(E i ) is regarded as a subgroup of i∈Λ G(E i ). Thus, for any (g i ) i∈Λ 0 ∈ i∈Λ 0 G(E i ) we obtain the product i∈Λ 0 g i ∈ G(∪ i∈Λ 0 E i ). When Λ 0 is a finite subset, the element i∈Λ 0 g i coincides with the usual product of g i 's in G, which is independent of the order of g i 's. Lemma 3.3. If G has a weak topology and has the multiplication supported by a discrete family {E i } i∈Λ of compact subsets in M , then the multiplication map η is continuous. Proof. Since η is a group homomorphism between topological groups, it suffices to show that the map η is continuous at the unit element e Λ = (e) i of i G(E i ). Given any neighborhood U of e in G, there exists a neighborhood V of e in G and a compact subset K of M such that V 2 ⊂ U and Since the finite multiplication map In this subsection we incorporate the arguments in the previous subsections and deduce a practical criterion, Proposition 3.1, which is used in Sections 4 and 5. Now consider the following data: (1) M is a space, {E i } i∈Λ is a discrete family of compact subsets in M and D i is a compact subset of E i for each i ∈ Λ. (2) G is a transformation group on M which has a weak topology and has the multiplication supported by the family {E i } i∈Λ . (3) (E, f 0 ) is a pointed space equipped with a support function on M and a continuous right action of G. Suppose that (i) supp f 0 = ∅ and (ii) supp f g ⊂ supp f ∪ supp g for any (f, g) ∈ E × G. (4) For each i ∈ Λ (a) (B i , α i ) is a pointed space and (b) P i : (E, f 0 ) → (B i , α i ) and G i : (B i , α i ) → (G(E i ), e) are pointed maps. Assume that these maps satisfy the following conditions. Assumption 3.1 yields the following conclusions; By (2) and Lemma 3.3 the multiplication map For simplicity, we use the symbol 4) are combined to yield the following maps between pointed pairs: These maps are defined by the formula: If f ∈ E c , then the compact set supp f meets only finitely many D i 's. Thus by (4)(i) P i (f ) = α i except finitely many i's and so The maps P , F and G determine two maps These maps are defined by Φ(f ) = (P (f ), F (f )) and Ψ(µ, h) = h • G(µ). tively. Then the homeomorphism Φ restricts to the homeomorphism of the subpairs Proof. (i) By Complement 3.1 it suffices to verify the following conditions: The condition ( * 1 ) follows from the definition of the map F . Since P (h) = α and G(µ) = G i (µ i ) on D i , by (4)(iii) it follows that P i (h • G(µ)) = µ i . This implies the condition ( * 2 ). (ii) (a) By the conditions on (E 1 , E 2 ) the map F induces the map between subpairs, F : Thus the assertion follows from (i) and Complement 3.1. (b) Since E c 1 is G c -invariant, the statement follows from (a). In this section we study the stability proeprty of diffeomorphism groups (Theorem 1.1 (1)) and prove Theorem 1.2 (1). Suppose M is a smooth (separable metrizable) n-manifold possibly with boundary. When M is orientable, the volume forms on M serves our purpose. However, to include the non-orientable case, it is necessary to recall the notion of volume density. Suppose V is a 1-dimensional real vector space. The dual space V * consists of all linear functions f : V → R, while its variant V # is defined by These spaces form 1-dimensional real vector spaces under the usual sum and scalor product of realvalued functions. When V is oriented, by V + we denote the connected component of V -{0} consisting of positive vectors. Even if V itself is not oriented, the space V # always admits a canonical orientation with the positive vectors In addition, if V is oriented, then V * admits the corresponding orientation and a canonical orientation-preserving isomorphism V * ∼ = V # . This construction extends to real line bundles. For any smooth real line bundle L → M , we have the associated real line bundles L * = ∪ p∈M (E p ) * → M and L # = ∪ p∈M (L p ) # → M . The line bundle L # has a canonical orientation, so that it is trivial since M is paracompact. If L itself is oriented (i.e., each fiber L p (p ∈ M ) is equipped with an orientation o p which varies continuously in p ∈ M ), then L * also admits the corresponding orientation and there exists a canonical isomorphism L * ∼ = L # of oriented vector bundles over M . Proof. The trivial fiber bundle L includes the sub-bundle L + = ∪ p∈M (L p ) + , which is also a trivial fiber bundle with fiber (0, ∞) ∼ = R. Since Γ + (L) = Γ(L + ) ∼ = C ∞ (M, R) and the latter is an infinitedimensional separable Frechet space, we have the conclusion. Now we apply the above arguments to the line bundle ∧ n T M . Any section ω ∈ Γ((∧ n T M ) # ) is called a density on M , since its components over coordinate charts transform by the absolute value of Jacobian under coordinate transitions and hence the integral M ω ∈ R is well-defined whenever ω has a compact support and the ω-volume ω(M ) = M ω ∈ (0, ∞] is defined as an improper integral for any positive density ω ∈ Γ + ((∧ n T M ) # ). To simplify the notations, let Suppose N is another smooth n-manifold possibly with boundary and f : Then the differential of f , df : This defines a continuous map It is seen that the group D(M ) acts continuously on the space V # (M ) by the pull-back and the subspace V # + (M ) is invariant under this action. For the inclusion i : N ⊂ M , the pull-back i * ω is also denoted by ω| N . For µ, ν ∈ V # (M ) we write µ ∼ 1 ν if ν = cµ for some c > 0. This is an equivalence relation and preserved by the pull-back. The group G acts continuously on the space E by the right composition, and as a transformation group on M it has a weak topology and also has the multiplication supported by the family {E i } i∈Λ . Hence, the composition map η : i∈Λ G(E i ) → G is continuous by Lemma 3.3. Since E = ∅, we can choose a distinguished element f 0 ∈ E. A support function for E on M is defined by Note that it satisfies the condition in Assumption 3.1 (3) and the subspace E c is G c -invariant (i.e., For each i ∈ Λ define a pointed space (B i , α i ) and two maps P i and G i as follows: Let The maps P i and G i are defined by Claim. The maps P i and G i satisfy the conditions (i) -(iii) in Assumption 3.1 (4). This implies that (f Hence, we can apply the arguments in Section 3.3 to this setting. Two pointed pairs (B, B c , α) and (F, F c , f 0 ) and three maps P , G and F are defined by These maps determine two maps Φ and Ψ by Proof. From Lemma 4.2 it follows that The next proposition follows from Proposition 3.1. ( restricts to the homeomorphism of the subpairs In particular, the pair then the map Φ induces the homeomorphism of the subpairs where Example 4.1. The space E includes the following G-invariant subspaces; For each i = 1, 2, 3, the map Φ induces the homeomorphism between the subpairs Φ : Thus, the pair Next we consider the case where M = N . As a base point of the space E we take f 0 = id M . Then the support function supp Then the map Φ induces the homeomorphism regard as s = (-1, 1) ∞ instead of R ∞ if necessary, and use the symbol (s Λ , s Λ f ) to denote the pair i∈Λ s, i∈Λ s ∼ = i∈Λ ℓ 2 , i∈Λ ℓ 2 for notational simplicity. 5.1. Morse's µ-length of arcs. Suppose (X, d) is a metric space and A is an arc in X. The arc A admits a canonical linear order ≤ unique up to the reversion. For each k ≥ 1 set The µ-length of A is defined by We use the following property of the quantity µ(A). Next we recall some basic facts on good Radon measures. A Radon measure on a space X is a Borel measure µ on X such that µ(K) < ∞ for any compact subset K of X. The measure µ is called good if µ(p) = 0 for any point p ∈ X and µ(U ) > 0 for any nonempty open subset U of X. Let M(X) denote the space of all Radon measures ν on X endowed with the weak topology. For µ, ν ∈ M(X) we say that ν is µ-biregular if µ(A) = 0 iff ν(A) = 0 for any Borel subset A of X. For µ ∈ M(X) and a Borel subset A of X let We need Oxtoby-Ulam theorem ( [17]) and Fathi's selection theorem ( [8]). A typical example of a good Radon measure is the Lubesgue measure m on R n (n ≥ 1). For any Proof. When n = 1, the assertion is obvious. Below we assume that n ≥ 2. (1) First we treat the case where (E, A) = (B n , [-1, 1]), θ = id [-1,1] and µ is m-biregular. Consider the decomposition of the n-cube, B n = B + ∪ B 0 B -, where We can easily find a map This completes the proof. Finally we deduce the (s ∞ , s ∞ f )-stability of groups of measure-preserving homeomorphisms. The next example includes Theorem 1.1 (3). Example 5.3. Suppose X is a subset of M -∪ i∈Λ Int E i and µ ∈ M ∪ i (A i ∪∂E i ) g (M ). We assume that n i ≥ 2 and µ i = µ| E i for each i ∈ Λ. Then the following pairs are (s Λ , s Λ f )-stable: (H X (M, µ), H c X (M, µ)), (H X (M, µ) i , H c X (M, µ) i ) (i = 0, 1) and (H X (M, µ) 1 , H c X (M, µ) * 1 ). This follows from Proposition 5.2, since (G, G c ) ⊂ (H X (M, µ) 1 , H c X (M, µ) * 1 ). Division of Mathematics, Graduate School of Science and Technology, Kyoto Institute of Technology, Matsugasaki, Sakyoku, Kyoto 606-8585, Japan E-mail address: yagasaki@kit.ac.jp