### A note on the homotopical characterization of Rn.

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A topological space X is called an ${\stackrel{\u02c7}{H}}^{n}$-bubble (n is a natural number, ${\stackrel{\u02c7}{H}}^{n}$ is Čech cohomology with integer coefficients) if its n-dimensional cohomology ${\stackrel{\u02c7}{H}}^{n}\left(X\right)$ is nontrivial and the n-dimensional cohomology of every proper subspace is trivial. The main results of our paper are: (1) Any compact metrizable ${\stackrel{\u02c7}{H}}^{n}$-bubble is locally connected; (2) There exists a 2-dimensional 2-acyclic compact metrizable ANR which does not contain any ${\stackrel{\u02c7}{H}}^{2}$-bubbles; and (3) Every n-acyclic finite-dimensional $L{\stackrel{\u02c7}{H}}^{n}$-trivial metrizable compactum...

A metric space (X,ϱ) satisfies the disjoint (0,n)-cells property provided for each point x ∈ X, any map f of the n-cell ${B}^{n}$ into X and for each ε > 0 there exist a point y ∈ X and a map $g:{B}^{n}\to X$ such that ϱ(x,y) < ε, $\widehat{\varrho}(f,g)<\epsilon $ and $y\notin g\left({B}^{n}\right)$. It is proved that each homogeneous locally compact ANR of dimension >2 has the disjoint (0,2)-cells property. If dimX = n > 0, X has the disjoint (0,n-1)-cells property and X is a locally compact $L{C}^{n-1}$-space then local homologies satisfy ${H}_{k}(X,X-x)=0$ for k < n and Hn(X,X-x) ≠ 0.

In this survey, we consider several questions pertaining to homeomorphisms, including criteria for their existence in certain circumstances, and obstructions to their existence.

The cell-like approximation theorem of R. D. Edwards characterizes the n-manifolds precisely as the resolvable ENR homology n-manifolds with the disjoint disks property for 5 ≤ n < ∞. Since no proof for the n = 5 case has ever been published, we provide the missing details about the proof of the cell-like approximation theorem in dimension 5.

The aim of this paper is to prove the generalized Schoenflies theorem for the class of absolute suspensions. The question whether the finite-dimensional absolute suspensions are homeomorphic to spheres remains open. Partial solution to this question was obtained in [Sz] and [Mi]. Morton Brown gave in [Br] an ingenious proof of the generalized Schoenflies theorem. Careful analysis of his proof reveals that modulo some technical adjustments a similar argument gives an analogous result for the class...