On the second homotopy group of SC(Z)SC(Z)

In our earlier paper (K. Eda, U. Karimov, and D. Repov\v{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 present note we establish some new algebraic properties of $SC(Z)$

From uncountable abelian groups to uncountable nonabelian groups

The present note surveys my research related to generalizing notions of abelian group theory to non-commutative case and applying them particularly to investigate fundamental groups

A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions

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 1950's

Finite sheeted covering maps over 2-dimensional connected, compact Abelian groups

AbstractLet G be a 2-dimensional connected, compact Abelian group and s be a positive integer. We prove that a classification of s-sheeted covering maps over G is reduced to a classification of s-index torsionfree supergroups of the Pontrjagin dual Gˆ. Using group theoretic results from earlier paper we demonstrate its consequences. We also prove that for a connected compact group Y:(1)Every finite-sheeted covering map from a connected space over Y is equivalent to a covering homomorphism from a compact, connected group.(2)If two finite-sheeted covering homomorphisms over Y are equivalent, then they are equivalent as topological homomorphisms

Finite index supergroups and subgroups of torsionfree abelian groups of rank two

AbstractEvery torsionfree abelian group A of rank two is a subgroup of Q⊕Q and is expressed by a direct limit of free abelian groups of rank two with lower diagonal integer-valued 2×2-matrices as the bonding maps. Using these direct systems we classify all subgroups of Q⊕Q which are finite index supergroups of A or finite index subgroups of A. Using this classification we prove that for each prime p there exists a torsionfree abelian group A satisfying the following, where A⩽Q⊕Q and all supergroups are subgroups of Q⊕Q:(1)for each natural number s there are ∑q|s,gcd(p,q)=1q s-index supergroups and also ∑q|s,gcd(p,q)=1q s-index subgroups;(2)each pair of distinct s-index supergroups are non-isomorphic and each pair of distinct s-index subgroups are non-isomorphic

On Snake cones, Alternating cones and related constructions

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 Repov\vs. We also introduce related constructions $CSC(-)$ and $CAC(-)$ and investigate homotopical differences between these four constructions. Finally, we explicitly describe the second homology group of the Hawaiian tori wedge