Let Y be a connected group and let f : X -> Y be a covering map with the total space X being connected: We consider the following question: Is it possible to define a topological group structure on X in such a way that f becomes a homomorphism of topological groups. This holds in some particular cases: if Y is a pathwise connected and locally pathwise connected group or if f is a finite-sheeted covering map over a compact connected group Y. However, using shape-theoretic techniques and Fox's notion of an overlay map, we answer the question in the negative. We consider infinite-sheeted covering maps over solenoids, i.e. compact connected 1-dimensional abelian groups. First we show that an infinite-sheeted covering map f : X -> Sigma with a total space being connected over a solenoid Sigma does not admit a topological group structure on X such that f becomes a homomorphism. Then, for an arbitrary solenoid Sigma, we construct a connected space X and an infinite-sheeted covering map f : X -> Sigma, which provides a negative answer to the question.

We show that the Snake on a square SC(S1) is homotopy equivalent to the space AC(S1) which was investigated in the previous work by Eda, Karimov and Repovš. 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.

We classify the groups that are the inverse limits of sequences of free groups of finite rank. Except free groups of finite rank, there exist four groups that are the inverse limits of sequences of free groups of finite rank. © 2013 London Mathematical Society.

We strengthen previous results on the fundamental groups of the Hawaiian earring and wild Peano continua. Let X be a path-connected, locally path-connected, first countable space which is not locally semi-simply connected at any point. If the fundamental group pi(1)(X) is a subgroup of a free product *H-j is an element of J(j), then it is contained in a conjugate subgroup to some H-j.

In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called 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, *) is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces SC (Z, *) for any path-connected compact spaces Z.

We prove that the one-point union of two copies of the cone over the Hawaiian earring is aspherical.

Let X and Y be one-dimensional Peano continua. If the fundamental groups of X and Y are isomorphic, then X and Y are homotopy equivalent. Every homomorphism from the fundamental group of X to that of Y is a composition of a homomorphism induced from a continuous map and a base point change isomorphism.

In our earlier paper we introduced a cone-like space SC(Z). In the present 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. (C) 2008 Elsevier B.V. All rights reserved.

Every torsionfree abelian group A of rank two is a subgroup of Q circle plus Q and is expressed by a direct limit of free abelian groups of rank two with lower diagonal integer-valued 2 x 2-matrices as the bonding maps. Using these direct systems we classify all subgroups of Q circle plus 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 circle plus Q and all supergroups are subgroups of Q circle plus Q:
(1) for each natural numbers there are Sigma(q vertical bar s),(gcd(p,q)=1) q s-index supergroups and also Sigma(q vertical bar s),(gcd(p,q)=1) s-index subgroups;
(2) each pair of distinct s-index supergroups are non-isomorphic and each pair of distinct s-index, subgroups are non-isonnorphic. (c) 2008 Elsevier Inc. All rights reserved.

It has been known for a long time that the fundamental group of the quotient of ℝ3 by the Case-Chamberlin continuum is nontrivial. In the present paper we prove that this group is in fact, uncountable.

Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts from a noncontractible n-dimensional Peano continuum for any n > 0, then our construction yields a simply connected noncontractible (n + l)-dimensional cell-like Peano continuum. In particular, starting from the circle S-1, one gets a noncontractible simply connected cell-like 2-dimensional Peano continuum.

Let 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 (G) over cap. 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. (C) 2005 Elsevier B.V. All rights reserved.

For each non-quadratic p-adic integer, p > 2, we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T-2 and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f(0): X-0 -> Y, f(1) : X-1 -> Y, f(2): X-2 -> Y such that the total spaces X-0 = Y, X-1 and X-2 are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-shected covering map f(3): X-3 -> Y such that X-3 and Y are nonhomeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps. (c) 2004 Elsevier B.V. All rights reserved.

Let X be a Peano continuum. Then the following hold: (1) The singular cohomology group H I (X) is isomorphic to the tech cohomology group h I (X). (2) For each homomorphism h : pi(1)(X) -> *(i is an element of l)G(i) there exists a finite subset F of I such that Im(h) subset of *(i is an element of F)G(i). (3) For each injective homomorphism h : pi(1) (X) -> G(0) * G(1) there exists a finitely generated subgroup F-0 of G(0) or a finitely generated subgroup F-1 of G(1) such that Im(h) subset of F-0 * G(1) or Im(h) subset of G(0) * F-1. (c) 2004 Published by Elsevier B.V.

We introduce a new construction of spaces from groups using homomorphic images of the fundamental group of the Hawaiian earring H. According to this construction the Menger sponge, Sierpinski gasket, Sierpinski carpet and the direct product of their countable many copies are recovered from their fundamental groups. (C) 2004 Published by Elsevier B.V

Let X be a one-dimensional metric space and ℍ be the Hawaiian earring.(1) Each homomorphism from π1(ℍ) to π1(X) is induced from a continuous map up to the base-point-change isomorphism on π1(X).(2) Let X be a one-dimensional Peano continuum. Then X has the same homotopy type as that of ℍ if and only if π1(X) is isomorphic to π1(ℍ), if and only if X has a unique point at which X is not semi-locally simply connected. (3) Let X and Y be one-dimensional Peano continua which are not semi-locally simply connected at any point. Then, X and Y are homeomorphic if and only if π1(X) and π1(Y) are isomorphic. Moreover, each isomorphism from π1(X) to π1(Y) is induced by a homeomorphism from X to Y up to the base-point-change-isomorphism. © 2001 Elsevier Science B.V. All rights reserved.

Let X be a one-dimensional metric space and H be the Hawaiian earring.
(1) Each homomorphism from pi(1) (H) to pi(1) (X) is induced from a continuous map up to the base-point-change isomorphism on pi(1) (X).
(2) Let X be a one-dimensional Peano continuum. Then X has the same homotopy type as that of H if and only if pi(1) (X) is isomorphic to pi(1) (H), if and only if X has a unique point at which X is not semi-locally simply connected.
(3) Let X and Y be one-dimensional Peano continua which are not semi-locally simply connected at any point. Then, X and Y are homeomorphic if and only if pi(1) (X) and pi(1) (Y) are isomorphic. Moreover, each isomorphism from pi(1) (X) to pi(1) (Y) is induced by a homeomorphism from X to Y up to the base-point-change-isomorphism. (C) 2001 Elsevier Science B.V. All rights reserved.

We prove that there exists a cohomology locally connected compact metrizable space which is not homology locally connected. In the category of compact Hausdorff spaces a similar result was proved earlier by G.E. Bredon. © 2002 Elsevier Science B.V. All rights reserved.

Let X be a one-dimensional space which contains a copy C of a circle and let it not be semi-locally simply connected at any point on C. Then the fundamental group of X cannot be embeddable into a free σ-product of n-slender groups, for instance, the fundamental group of the Hawaiian earring. Consequently, any one of the fundamental groups of the Sierpinski gasket, the Sierpinski curve, and the Menger curve is not embeddable into the fundamental group of the Hawaiian earring.

Let X be a one-dimensional space which contains a copy C of a circle and let it not be semi-locally simply connected at any point on C: Then the fundamental group of X cannot be embeddable into a free sigma-product of n-slender groups, for instance, the fundamental group of the Hawaiian earring. Consequently, any one of the fundamental groups of the Sierpinski gasket, the Sierpinski curve, and the Menger curve is not embeddable into the fundamental group of the Hawaiian earring.

Let X be a locally n-connectcd compact metric space. Then, the canonical homomorphism from the singular homology group Hn+1(X) to the Čech homology group Ȟn+1(X) is surjective. Consequently, if a compact metric space X is locally connected, then the canonical homomorphism from H1 (X) to Ȟ1 (X) is surjective. ©2000 American Mathematical Society.

For the n-dimensional Hawaiian earring ℍn, n ≥ 2, πn(ℍn, o) ≃ ℤω and πi(ℍn,o) is trivial for each 1 ≤ i ≤ n - 1. Let CX be the cone over a space X and CXVCY be the one-point union with two points of the base spaces X and Y being identified to a point. Then Hn(X V Y) ≃ Hn(X) ⊕ Hn(Y) ⊕ Hn(CX V CY) for n ≥ 1.

Let X be a space of dimension at most 1. Then, the fundamental group is isomorphic to a subgroup of the first Čech homotopy group based on finite open covers. Consequently, for a one-dimensional continuum X, the fundamental group is isomorphic to a subgroup of the first Čech homotopy group. © 1998 Elsevier Science B.V.