The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $mathbb{D}^n$; (2) Every $n$-dimens
ional cell-like compactum can be embedded into $(2n+1)$-dimensional Euclidean space as a cellular subset; and (3) There exists a locally compact planar set which is acyclic with respect to v{C}ech homology and whose fine coverings are all nonacyclic.
We show that the Snake on a square $SC(S^1)$ is homotopy equivalent to the space $AC(S^1)$ which was investigated in the previous work by Eda, Karimov and Repovvs. We also introduce related constructions $CSC(-)$ and $CAC(-)$ and investigate homotopi
cal differences between these four constructions. Finally, we explicitly describe the second homology group of the Hawaiian tori wedge.
We construct a functor $AC(-,-)$ from the category of path connected spaces $X$ with a base point $x$ to the category of simply connected spaces. The following are the main results of the paper: (i) If $X$ is a Peano continuum then $AC(X,x)$ is a cel
l-like Peano continuum; (ii) If $X$ is $n-$dimensional then $AC(X, x)$ is $(n+1)-$dimensional; and (iii) For a path connected space $X$, $pi_1(X,x)$ is trivial if and only if $pi_2(AC(X, x))$ is trivial. As a corollary, $AC(S^1, x)$ is a 2-dimensional nonaspherical cell-like Peano continuum.
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.
We calculate the singular homology and v{C}ech cohomology groups of the Harmonic archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Edas proof that the first singular homology groups of these spaces are isomorphic.
In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called {sl Snake space}. In the sequel we introduced the functor $SC(-,-)$ defined on the category of all spaces with base points
and continuous mappings. For the circle $S^1$, the space $SC(S^1, ast)$ is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces $SC(Z, ast)$ for any path-connected compact spaces $Z$.
We construct an example of a Peano continuum $X$ such that: (i) $X$ is a one-point compactification of a polyhedron; (ii) $X$ is weakly homotopy equivalent to a point (i.e. $pi_n(X)$ is trivial for all $n geq 0$); (iii) $X$ is noncontractible; and (i
v) $X$ is homologically and cohomologically locally connected (i.e. $X$ is a $HLC$ and $clc$ space). We also prove that all classical homology groups (singular, v{C}ech, and Borel-Moore), all classical cohomology groups (singular and v{C}ech), and all finite-dimensional Hawaiian groups of $X$ are trivial.
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 p
resent note we establish some new algebraic properties of $SC(Z)$.
We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths space from the 1950s.
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 $n ge 3$.
