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