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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}.
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We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of the product of the two-sphere and the circle. This characterization can also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group. We also determine which three-manifolds are dominated by non-trivial circle bundles, and which three-manifold groups are presentable by products.
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}.
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 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.
In our earlier paper (K. Eda, U. Karimov, and D. Repovv{s}, emph{A construction of simply connected noncontractible cell-like two-dimensional Peano continua}, Fund. Math. textbf{195} (2007), 193--203) we introduced a cone-like space $SC(Z)$. In the present note we establish some new algebraic properties of $SC(Z)$.
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