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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.
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We show that the classical example $X$ of a 3-dimensional generalized manifold constructed by van Kampen is another example of not homologically locally connected (i.e. not HLC) space. This space $X$ is not locally homeomorphic to any of the compact metrizable 3-dimensional manifolds constructed in our earlier paper which are not HLC spaces either.
Suppose that $X=G/K$ is the quotient of a locally compact group by a closed subgroup. If $X$ is locally contractible and connected, we prove that $X$ is a manifold. If the $G$-action is faithful, then $G$ is a Lie group.
We show that a homeomorphism of a semi-locally connected compact metric space is equicontinuous if and only if the distance between the iterates of a given point and a given subcontinuum (not containing that point) is bounded away from zero. This is false for general compact metric spaces. Moreover, homeomorphisms for which the conclusion of this result holds satisfy that the set of automorphic points contains those points where the space is not semi-locally connected.
Let $P$ be a polynomial with a connected Julia set $J$. We use continuum theory to show that it admits a emph{finest monotone map $ph$ onto a locally connected continuum $J_{sim_P}$}, i.e. a monotone map $ph:Jto J_{sim_P}$ such that for any other monotone map $psi:Jto J$ there exists a monotone map $h$ with $psi=hcirc ph$. Then we extend $ph$ onto the complex plane $C$ (keeping the same notation) and show that $ph$ monotonically semiconjugates $P|_{C}$ to a emph{topological polynomial $g:Cto C$}. If $P$ does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwis fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a criterion and a useful sufficient condition for the map $ph$ not to collapse $J$ into a point.
Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions $n ge 3$.
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