On ${\displaystyle check{\left\{H\right\}}^n}$-bubbles in n-dimensional compacta

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