Embeddings of infinite-dimensional manifold pairs and remarks on stability and deficiency

In this paper, we treat of an E-manifold pair (M, N) with N a Z-set in M where E is an infinite-dimensional locally convex linear metric space which is homeomorphic to Eω or E[?]. And we study the condition under which M can be embedded in E such that N is the topological boundary under the embedding (Anderson\u27s Problem in [2]). Moreover we extend the results on topological stability and deficiency, the Homeomorphism Extension Theorem and the results in [18]. For each space X, we denote by Xω the coutable infinite product of X by itself. And for each space X with a base point 0, X[?]={(xi)εXωxi=0 for almost all i}. A closed subset K of a space X is a Z-set in X if for each non-empty homotopically trivial open set U, U[?]K is also non-empty and homotopically trivial([1]). An E-manifold is a paracompact manifold modelled on a space E. As a modelled space, ･･･Thesis--University of Tsukuba, D.Sc.(B), no. 20, 1979. 10. 3

Proper n-shape categories

In this paper, it is shown that the proper n-shape category of Ball-Sher type is isomorphic to a subcategory of the proper n-shape category defined by proper n-shaping. It is known that the latter is isomorphic to the shape category defined by the pair (Hpn, Hpn Pol), where Hpn is the category whose objects are locally compact separable metrizable spaces and whose morphisms are proper n-homotopy classes of proper maps, and Hpn Pol is the full subcategory of Hpn whose objects are spaces having the proper n-homotopy type of polyhedra. In the case n = ∞, this shows the relation between the original Ball-Sher\u27s category and the proper shape category defined by proper shapings. We also discuss the proper n-shape category of space of dimension ≤ n + 1

Combinatorial infinite-dimensional manifolds and R∞-manifolds

AbstractGeneralizing combinatorial manifolds to the infinite-dimensional case, we can define combinatorial ∞-manifolds. And then we can see that each R∞-manifold is triangulated by a combinatorial ∞-manifold. The main purpose of this paper is to prove the Hauptvermutung for combinatorial ∞-manifolds: Any two homeomorphic combinatorial ∞-manifolds are combinatorially equivalent, that is, they have simplicially isomorphic subdivisions. As an application, we have the stable Hauptvermutung for simplicial complexes: For any homeomorphic countable simplicial complexes K and L, the product complexes K × Δ∞ and L × Δ∞ are combinatorially equivalent, where Δ∞ is the countably infinite full complex

Homeomorphism and diffeomorphism groups of non-compact manifolds with the Whitney topology

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.Comment: 21 pages, Sections 3, 5.2, 5.3, 6, 7 in the previous version (arXiv: 0802.0337v1) will be included in another paper cited as Ref. [6

Describing the proper n-shape category by using non-continuous functions

In this paper, we describe the proper n-shape category by using non-continuous functions. Moreover, applying non-continuous homotopies, we show that the Cech expansion is a polyhedral expansion in the proper n-homotopy category

Correcting Taylor\u27s cell-like map

J. L. Taylor constructed a cell-like map of a compactum X onto the Hilbert cube IN such that X is not cell-like. In this note, we point out a defect in the construction and show how to fix it