Tatsuhiko Yagasaki

Homotopy types of homeomorphism groups of noncompact 2-manifolds

Abstract Suppose M is a noncompact connected PL 2-manifold and let H (M) 0 denote the identity component of the homeomorphism group of M with the compact-open topology. In this paper we classify the homotopy type of H (M) 0 by showing that H (M) 0 has the homotopy type of the circle if M is the plane, an open or half open annulus, or the punctured projective plane. In all other cases we show that H (M) 0 is homotopically trivial.

Spaces of embeddings of compact polyhedra into 2-manifolds

Let M be a PL 2-manifold and X be a compact subpolyhedron of M and let E(X,M) denote the space of embeddings of X into M with the compact-open topology. In this paper we study an extension property of embeddings of X into M and show that the restriction map from the homeomorphism group of M to E(X,M) is a principal bundle. As an application we show that if M is a Euclidean PL 2 -manifold and dimX≥1 then the triple (E(X,M) , ELIP(X,M) , EPL(X,M)) is an (s,Σ,σ) -manifold, where EKLIP(X,M) and EKPL(X,M) denote the subspaces of Lipschitz and PL embeddings.

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

Abstract We prove that a topological group G is (locally) homeomorphic to an LF-space if G = ⋃ n ∈ ω G n for some increasing sequence of subgroups ( G n ) n ∈ ω such that (1) for any neighborhoods U n ⊂ G n , n ∈ ω , of the neutral element e ∈ G n ⊂ G , the set ⋃ n = 1 ∞ U 0 U 1 ⋯ U n is a neighborhood of e in G ; (2) each group G n is (locally) homeomorphic to a Hilbert space; (3) for every n ∈ N the quotient map G n → G n / G n − 1 is a locally trivial bundle; (4) for infinitely many numbers n ∈ N each Z -point in the quotient space G n / G n − 1 = { x G n − 1 : x ∈ G n } is a strong Z -point.

Diffeomorphism groups of non-compact manifolds endowed with the Whitney C∞-topology

For every functional functor F : Comp → Comp in the category Comp of compact Hausdorff spaces we define the notions of F -Dugundji and F -Milutin spaces, generalizing the classical notions of a Dugundji and Milutin spaces. We prove that the class of F -Dugundji spaces coincides with the class of absolute F -valued retracts. Next, we show that for a monomorphic continuous functor F : Comp → Comp admitting tensor products each Dugundji compact is an absolute F -valued retract if and only if the doubleton {0, 1} is an absolute F -valued retract if and only if some points a ∈ F ({0}) ⊂ F ({0, 1}) and b ∈ F ({1}) ⊂ F ({0, 1}) can be linked by a continuous path in F ({0, 1}). We prove that for the functor Lipk of k-Lipschitz functionals with k < 2, each absolute Lipk-valued retract is openly generated. On the other hand the one-point compactification of any uncountable discrete space is not openly generated but is an absolute Lip3-valued retract. More generally, each hereditarily paracompact scattered compact space X of finite scattered height n = ht(X) is an absolute Lipk-valued retract for k = 2 n+2 − 1.

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

Abstract We prove that for any non-compact connected surface M the group H c ( M ) of compactly supported homeomorphisms of M endowed with the Whitney topology is homeomorphic to R ∞ × l 2 or Z × R ∞ × l 2 .

Infinite-dimensional manifold triples of homeomorphism groups

Abstract In this paper we give some characterizations of (s, Σ, σ)-, (s2, s × σ, σ2)- and (s∞, σ∞, σf∞)-manifold triples under the stability condition. As an application we show that if M is a compact PL n-manifold (n ⩾ 1, n ≠ 4 and ∂M = θ for n = 5) and H(M) is an ANR, then ( H(M) ∗ , H LIP (M) ∗ , H PL (M)) is an (s, Σ, σ)-manifold triple, where H(M) (HLIP(M) or HPL(M)) is the group of (Lipschitz or PL) homeomorphisms of M with the compact-open topology and H(M) ∗ (H (LIP) (M) ∗ ) is the subgroup consisting of (Lipschitz) homeomorphisms approximated by PL-homeomorphisms. We also show that a triple of homeomorphism groups of the real line is an (s∞, σ∞, σf∞)-manifold triple.

Embedding spaces and hyperspaces of polyhedra in 2-manifolds

Abstract Suppose M is a 2-manifold and X is a compact polyhedron. Let E (X,M) denote the space of embeddings of X into M with the compact-open topology and let K (X,M) denote the hyperspace of copies of X embedded in M with the Frechet topology. In this paper we show that the natural map π : E (X,M)→ K (X,M) , π(f)=f(X), is a principal bundle with fiber H (X) , the homeomorphism group of X. As a corollary it follows that the space K (X,M) is an ANR.

Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends

Suppose M is a noncompact connected oriented C ∞ n-manifold and ω 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 ∞ topology and let D(M; w) denote the subgroup of ω-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 Mosers theorem, enables us to deduce two selection theorems for the groups D + (M) and D(M;ω). The first one is the extension of Mosers 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; w) is a strong deformation retract of the group D + (M;E ω M ) consisting of h ∈ D + (M), which preserves the set E ω M of ω-finite ends of M. The second one is related to the mass flow toward ends under volume-preserving diffeomorphisms of M. Let D EM (M; w) denote the subgroup consisting of all h ∈ D(M; ω) which fix the ends E M of M. S. R. Alpern and V. S. Prasad introduced the topological vector space S(M; ω) of end charges of M and the end charge homomorphism c ω : D EM (M; ω) → S(M; ω), which measures the mass flow toward ends induced by each h ∈ D EM (M; ω). We show that the homomorphism c ω has a continuous section. This induces the factorization D EM (M; ω) ≅ ker c ω x S(M; ω), and it implies that ker c ω is a strong deformation retract of D EM (M; ω).

Hyperspaces of Peano and ANR continua

Abstract Suppose X is a locally connected continuum without free arcs. It is known that the hyperspace C(X) is homeomorphic to the Hilbert cube Q . Let L c (X) and ANR c (X) denote the subspaces of C(X) consisting of locally connected continua and ANR continua in X . In this article we show that if X has DD 1 P then the pair (C(X),L c (X)) is homeomorphic to (Q,Q 0 ) and if, in addition, every nonempty open subset of X contains a 3-disk then the pair (C(X), ANR c (X)) is homeomorphic to (Q,Ω 3 ) .

The groups of quasiconformal homeomorphisms on Riemann surfaces

Suppose M is a connected Riemann surface. Let H(M) denote the homeomorphism group of M with the compact-open topology, and HQC(M) denote the subgroup of quasiconformal mappings of M onto itself, and let H(M)0 and HQC(M)0 denote the identity components of H(M) and HQC(M) respectively. In this paper we show that the pair (H(M)0,HQC(M)0) is an (s, Σ)-manifold, and determine their topological types.