arXiv subjects
Publications and source records attributed to Tatsuhiko Yagasaki.
In this note we clarify general properties of the Hausdorff-like metric on the power set ${\cal S}(G)$ of a group $G$ induced from word length norm and obtain some results on quasi-isometries between some subspaces of ${\cal S}(G)$ and ${\cal S}(H)$ for a group epimorphism $f : G \to H$, when ${\cal S}({\rm Ker}\,f)$ is metrically bounded.
In this paper we study boundedness of conjugation invariant norms on diffeomorphism groups of manifold pairs. For the diffeomorphism group ${\mathcal D} \equiv {\rm Diff}(M,N)_0$ of a closed manifold pair $(M, N)$ with $\dim N \geq 1$, first we clarify the relation among the fragmentation norm, the conjugation generated norm, the commutator length $cl$ and the commutator length with support in balls $clb$ and show that ${\mathcal D}$ is weakly simple relative to a union of some normal subgroups of ${\mathcal D}$. For the boundedness of these norms, this paper focuses on the case where $N$ is a union of $m$ circles. In this case, the rotation angle on $N$ induces a quasimorphism $ν: {\rm Isot}(M, N)_0 \to {\Bbb R}^m$, which determines a subgroup $A$ of ${\Bbb Z}^m$ and a function $\widehatν : {\mathcal D} \to {\Bbb R}^m/A$. If ${\rm rank}\,A = m$, these data leads to an upper bound of $clb$ on ${\mathcal D}$ modulo the normal subgroup ${\mathcal G} \cong {\rm Diff}_c(M - N)_0$. Then, some upper bounds of $cl$ and $clb$ on ${\mathcal D}$ are obtained from those on ${\mathcal G}$. As a consequence, the group ${\mathcal D}$ is uniformly weakly simple and bounded when $\dim M \neq 2,4$. On the other hand, if ${\rm rank}\,A < m$, then the group ${\mathcal D}$ admits a surjective quasimorphism, so it is unbounded and not uniformly perfect. We examine the group $A$ in some explicit examples.
In this paper we study boundedness of bundle diffeomorphism groups over a circle. For a fiber bundle $π: M \to S^1$ with fiber $N$ and structure group $Γ$ and $r \in {\Bbb Z}_{\geq 0} \cup \{ \infty \}$ we distinguish an integer $k = k(π, r) \in {\Bbb Z}_{\geq 0}$ and construct a function $\widehatν : {\rm Diff}_π(M)_0 \to {\Bbb R}_k$. When $k \geq 1$, it is shown that the bundle diffeomorphism group ${\rm Diff}_π(M)_0$ is uniformly perfect and $clb_π\,{\rm Diff}^r_π(M)_0 \leq k+3$, if ${\rm Diff}_{ρ, c}(E)_0$ is perfect for the trivial fiber bundle $ρ: E \to {\Bbb R}$ with fiber $N$ and structure group $Γ$. On the other hand, when $k = 0$, it is shown that $\widehatν$ is a unbounded quasimorphism, so that ${\rm Diff}_π(M)_0$ is unbounded and not uniformly perfect. We also describe the integer $k$ in term of the attaching map $ϕ$ for a mapping torus $π: M_ϕ\to S^1$ and give some explicit examples of (un)bounded groups.
In this paper we study the uniform perfectness, boundedness and uniform simplicity of diffeomorphism groups of compact manifolds with boundary and open manifolds and obtain some upper bounds of their diameters with respect to commutator length, those with support in balls and conjugation-generated norm.
A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ\leq 0 the κ-cone ends over any compact Lipschitz metric manifolds, all of them have the property (LD). We also formulate a notion of end deformation property for uniform embeddings over proper product ends (ED). For example, the 0-cone end over a compact metric manifold has the property (ED) if it has the property (LD). It is shown that if a metric manifold M has finitely many proper product ends with the property (ED), then the group of bounded uniform homeomorphisms of M endowed with the uniform topology admits a strong deformation retraction onto the subgroup of bounded uniform homeomorphisms which are identity over those ends. We also study a role of uniform isotopies in deformation of uniform homeomorphisms and show that Alexander isotopies in κ-cones induce contractions of some subgroups of groups of bounded uniform homeomorphisms.
In this paper we deduce a local deformation lemma for uniform embeddings in a metric covering space over a compact manifold from the deformation lemma for embeddings of a compact subspace in a manifold. This implies the local contractibility of the group of uniform homeomorphisms of such a metric covering space under the uniform topology. Further more, combining with similarity transformation, this enables us to induce a global homotopy property of groups of uniform homeomorphisms of metric spaces with Euclidean ends. In particular, we show that the identity component of the group of uniform homeomorphisms of the standard Euclidean n-space is contractible.
Suppose M is a non-compact connected n-manifold without boundary, DD(M) is the group of C^\infty-diffeomorphisms of M endowed with the Whitney C^\infty-topology and DD_0(M) is the identity connected component of DD(M), which is an open subgroup in the group DD_c(M) \subset DD(M) of compactly supported diffeomorphisms of M. It is shown that DD_0(M) is homeomorphic to N \times IR^\infty for an l_2-manifold N whose topological type is uniquely determined by the homotopy type of DD_0(M). For instance, DD_0(M) is homeomorphic to l_2 \times IR^\infty if n = 1, 2 or n = 3 and M is orientable and irreducible. We also show that for any compact connected n-manifold N with non-empty boundary \partial N the group DD_0(N - \partial N) is homeomorphic to DD_0(N; \partial N) \times IR^\infty, where DD_0(N;\partial N) is the identity component of the group DD(N;\partial N) of diffeomorphisms of N that do not move points of the boundary \partial N.
We prove that for any non-compact connected surface $M$ the group $H_c(M)$ of compactly suported homeomorphisms of $M$ endowed with the Whitney topology is homeomorphic to $R^\infty\times l_2$ or $Z\times R^\infty\times l_2$.
Suppose M is a non-compact connected smooth n-manifold. Let D(M) denote the group of diffeomorphisms of M endowed with the compact-open C^\infty-topology and D^c(M) denote the subgroup consisting of diffeomorphisms of M with compact support. Let D(M)_0 and D^c(M)_0 be the connected components of id_M in D(M) and D^c(M) respectively. In this paper we show that the pair (D(M), D^c(M)) admits a topological (\prod^ω\ell_2, \sum^ω\ell_2)-factor. In the case n = 2, this enables us to apply the characterization of (\prod^ω\ell_2, \sum^ω\ell_2)-manifolds and show that the pair (D(M)_0, D^c(M)_0) is a (\prod^ω\ell_2, \sum^ω\ell_2)-manifold and determine its topological type. We also obtain a similar result for groups of homeomorphisms of non-compact topological 2-manifolds.
For a non-compact n-manifold M let H(M) denote the group of homeomorphisms of M endowed with the Whitney topology and H_c(M) the subgroup of H(M) consisting of homeomorphisms with compact support. It is shown that the group H_c(M) is locally contractible and the identity component H_0(M) of H(M) is an open normal subgroup in H_c(M). This induces the topological factorization H_c(M) \approx H_0(M) \times \M_c(M) for the mapping class group \M_c(M) = H_c(M)/H_0(M) with the discrete topology. Furthermore, for any non-compact surface M, the pair (H(M), H_c(M)) is locally homeomorphic to (\square^w l_2,\cbox^w l_2) at the identity id_M of M. Thus the group H_c(M) is an (l_2 \times R^\infty)-manifold. We also study topological properties of the group D(M) of diffeomorphisms of a non-compact smooth n-manifold M endowed with the Whitney C^\infty-topology and the subgroup D_c(M) of D(M) consisting of all diffeomorphisms with compact support. It is shown that the pair (D(M),D_c(M)) is locally homeomorphic to (\square^w l_2, \cbox^w l_2) at the identity id_M of M. Hence the group D_c(M) is a topological (l_2 \times R^\infty)-manifold for any dimension n.
Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps f:X \to G endowed with the Whitney (graph) topology and by C_c(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X,G) is an l_2-manifold. In this article we show that if X is non-compact and not end-discrete then C_c(X,G) is an (R^\infty \times l_2)-manifold, and moreover the pair (C(X,G), C_c(X,G)) is locally homeomorphic to the pair of the box and the small box powers of l_2.
Suppose M is a noncompact connected smooth 2-manifold without boundary and let D(M)_0 denote the identity component of the diffeomorphism group of M with the compact-open C^infty-topology. In this paper we investigate the topological type of D(M)_0 and show that D(M)_0 is a topological ell_2-manifold and it has the homotopy type of the circle if M is the plane, the open annulus or the open M"obius band, and it is contractible in all other cases. When M admits a volume form w, we also discuss the topological type of the group of w-preserving diffeomorphisms of M. To obtain these results we study some fundamental properties of transformation groups on noncompact spaces endowed with weak topology.
We prove that the group D^r(R) of C^r diffeomorphisms of the real line, endowed with the compact-open and Whitney C^r topologies, is bihomeomorphic to the group H(R) of homeomorphisms of the real line endowed with the compact-open and Whitney topologies. This implies that the diffeomorphism group D^r(R) endowed with the Whitney C^r topology is homeomorphic to the countable box-power of the separable Hilbert space.
Suppose M is a noncompact connected oriented C^infty n-manifold and omega is a positive volume form on M. Let D^+(M) denote the group of orientation preserving diffeomorphisms of M endowed with the compact-open C^infty topology and D(M; omega) denote the subgroup of omega-preserving diffeomorphisms of M. In this paper we propose a unified approach for realization of mass transfer toward ends by diffeomorphisms of M. This argument together with Moser's theorem enables us to deduce two selection theorems for the groups D^+(M) and D(M; omega). The first one is the extension of Moser's theorem to noncompact manifolds, that is, the existence of sections for the orbit maps under the action of D^+(M) on the space of volume forms. This implies that D(M; omega) is a strong deformation retract of the group D^+(M; E^omega_M) consisting of h in D^+(M) which preserves the set E^omega_M of omega-finite ends of M. The second one is related to the mass flow toward ends under volume-preserving diffeomorphisms of M. Let D_{E_M}(M; omega) denote the subgroup consisting of all h in D(M; omega) which fix the ends E_M of M. S.R.Alpern and V.S.Prasad introduced the topological vector space S(M; omega) of end charges of M and the end charge homomorphism c^omega : D_{E_M}(M; omega) to S(M; omega), which measures the mass flow toward ends induced by each h in D_{E_M}(M; omega). We show that the homomorphism c^omega has a continuous section. This induces the factorization D_{E_M}(M; omega) cong ker c^omega times S(M; omega) and it implies that ker c^omega is a strong deformation retract of D_{E_M}(M; omega).
In this paper we introduce a general notion of weak extension property for embeddings induced by a group actions. As an example, for the group H(M, m) of measure-preserving homeomorphisms of a noncompact manifold M, we deduce weak type extension theorems, and as an application we exhibit the local contractibility of the group H_c(M, m) of measure-preserving homeomorphisms with compact support of a noncompact connected manifold M endowed with the Whitney topology.
Suppose M is a noncompact connected n-manifold and m is a good Radon measure of M with m(bdry M) = 0. Let H(M; m) denote the group of m-preserving homeomorphisms of M equipped with the compact-open topology and H_E(M; m) denote the subgroup consisting of all h in H(M; m) which fix the ends of M. Each h in H_E(M; m) moves mass toward ends and this quantity is measured by a mass flow homomorphism J : H_E(M; m) -> V_m, where V_m is a topological vector space. We show that the map J has a continuous section. This induces the factorization H_E(M; m) cong Ker J times V_m and implies that Ker J is a strong deformation retract of H_E(M; m).
Suppose M is a noncompact connected 2-manifold and m is a good Radon measure of M with m(partial M) = 0. Let H(M)_0 denote the identity component of the group of homeomorphisms of M equipped with the compact-open topology and let H(M; m)_0 denote the identity component of the subgroup consisting of m-preserving homeomorphisms of M. We use results of A.Fathi and R.Berlanga to show that H(M; m)_0 is a strong deformation retract of H(M)_0 and classify the topological type of H(M; m)_0.
Suppose M is a connected PL 2-manifold and X is a compact connected subpolyhedron of M (X \neq 1pt, a closed 2-manifold). Let E(X, M) denote the space of topological embeddings of X into M with the compact-open topology and let E(X, M)_0 denote the connected component of the inclusion i_X : X \subset M in E(X, M). In this paper we classify the homotopy type of E(X, M)_0 in term of the subgroup G = Im[{i_X}_\ast : π_1(X) \to π_1(M)]. We show that if G is not a cyclic group and M \neq T^2, T^2 then E(X, M)_0 \simeq \ast, if G is a nontrivial cyclic group and M \neq P^2, T^2, K^2 then E(X, M)_0 \simeq S^1, and when G = 1, if X is an arc or M is orientable then E(X, M)_0 \simeq ST(M) and if X is not an arc and M is nonorientable then E(X, M)_0 \simeq ST(\tilde{M}). Here S^1 is the circle, T^2 is the torus, P^2 is the projective plane and K^2 is the Klein bottle. The symbol ST(M) denotes the tangent unit circle bundle of M with respect to any Riemannian metric of M and \tilde{M} denotes the orientation double cover of M.