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Register a new userHomogeneous ANR-spaces and Alexandroff manifolds
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V. Valov
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We specify a result of Yokoi cite{yo} by proving that if $G$ is an abelian group and $X$ is a homogeneous metric $ANR$ compactum with $dim_GX=n$ and $check{H}^n(X;G) eq 0$, then $X$ is an $(n,G)$-bubble. This implies that any such space $X$ has the following properties: $check{H}^{n-1}(A;G) eq 0$ for every closed separator $A$ of $X$, and $X$ is an Alexandroff manifold with respect to the class $D^{n-2}_G$ of all spaces of dimension $dim_Gleq n-2$. We also prove that if $X$ is a homogeneous metric continuum with $check{H}^n(X;G) eq 0$, then $check{H}^{n-1}(C;G) eq 0$ for any partition $C$ of $X$ such that $dim_GCleq n-1$. The last provides a partial answer to a question of Kallipoliti and Papasoglu cite{kp}.
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We prove a homological characterization of $Q$-manifolds bundles over $C$-spaces. This provides a partial answer to Question QM22 from cite{w}.
We provide some properties and characterizations of homologically $UV^n$-maps and $lc^n_G$-spaces. We show that there is a parallel between recently introduced by Cauty algebraic $ANR$s and homologically $lc^n_G$-metric spaces, and this parallel is similar to the parallel between ordinary $ANR$s and $LC^n$-metric spaces. We also show that there is a similarity between the properties of $LC^n$-spaces and $lc^n_G$-spaces. Some open questions are raised.
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.
We investigate the classical Alexandroff-Borsuk problem in the category of non-triangulable manifolds: Given an $n$-dimensional compact non-triangulable manifold $M^n$ and $varepsilon > 0$, does there exist an $varepsilon$-map of $M^n$ onto an $n$-dimensional finite polyhedron which induces a homotopy equivalence?
The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $mathbb{D}^n$; (2) Every $n$-dimensional cell-like compactum can be embedded into $(2n+1)$-dimensional Euclidean space as a cellular subset; and (3) There exists a locally compact planar set which is acyclic with respect to v{C}ech homology and whose fine coverings are all nonacyclic.
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