The first singular homology of a Peano continuum X with torsion-free first Cech homology, Ȟ_1(x), splits as H_1(X) = Ȟ_1(X) ⊕ K where K is the homology shape kernel of X. Consequently if a Peano continuum X is a subspace of ℝ^3, then H_1(X) = ℤ^λ⊕ K where K is the homology shape kernel of X and λ is a countable cardinal. In the process we construct cotorsion quotients of subgroups of the first homology which correspond to path-connected fibrations of X.

The notions of tree-like loop and Lipschitz tree-like loop were introduced by Hambly and Lyons in their 2010 Annals of Mathematics paper. They showed that the Lipschitz tree-like property determines an equivalence relation on the set of paths of bounded variation in a given metric space and then asked if this notion could be extended to paths without the Lipschitz requirement. We show that after eliminating the Lipschitz requirement, the resulting relation is no longer transitive and thus is not an equivalence relation. The counterexample is obtained by analyzing an explicit fractal construction in the plane.

We prove a general theorem giving constraints on maps from certain topological groups to inverse limits of bounded torsion groups. From this we obtain some automatic continuity and ultraproduct results. For example, every homomorphism from a Polish group to a countable torsion-free residually finite group has open kernel. Also, the Grigorchuk group is a homomorphic image of a nonprincipal ultraproduct of groups if and only if there exists a measurable cardinal.

A map p: E-X has the unique path lifting property if every path in X, after a choice of an initial point, lifts uniquely to a path in E. We prove that if a group G acts on an ft8-tree T in such a way that the quotient map p: T-T/G has the unique path lifting property, then the quotient space T/G does not contain a disc. As a consequence, we show that every map of manifolds with the unique path lifting property is a covering map. The proof requires a study of one-dimensional backtracking in paths. We show the surprising and counterintuitive result that the equivalence relation given by homotopies of paths rel. endpoints is generated by inserting and deleting one-dimensional backtracking.

Let X be a Peano continuum (i.e., a metric space that is compact, connected and locally connected). We show that every path-connected inverse limit of covering spaces over X is determined by its fundamental group and is homeomorphic to a quotient of the set of homotopy classes of based paths endowed with the shape topology.

Suppose a path $\alpha$ factors through an $\mathbb{R}$-tree $T$ as $\alpha=q \circ p $. Let $r$ parameterize the unique geodesic in $T$ joining the endpoints of $p$. Then we say that the path $\beta=q \circ r$ is obtained from $\alpha$ by "geodesic $\mathbb{R}$-tree reduction." Essentially, $\beta$ is obtained from $\alpha$ by deleting one-dimensional back-tracking. In this paper, we show that any two homotopic paths are geodesic $\mathbb{R}$-tree reductions of some single common path. Hence, the equivalence relation on paths generated by geodesic $\mathbb{R}$-tree reduction is precisely path-homotopy. The common path is explicitly constructed and is necessarily space-filling in the image of a given path-homotopy.

Classically, an abelian group $G$ is said to be slender if every homomorphism from the countable product $\mathbb Z^{\mathbb N}$ to $G$ factors through the projection to some finite product $\mathbb Z^n$. Various authors have proposed generalizations to non-commutative groups, resulting in a plethora of similar but not completely equivalent concepts. In the first part of this work we present a unified treatment of these concepts and examine how are they related. In the second part of the paper we study slender groups in the context of co-small objects in certain categories, and give several new applications including the proof that certain homology groups of Barratt-Milnor spaces are cotorsion groups and a universal coefficients theorem for \v{C}ech cohomology with coefficients in a slender group.

We define the Peano dimension for groups arising as fundamental groups, which generalizes the classical definition of geometric dimension of finitely presented groups. We conjecture that the Peano dimension of the fundamental group of a aspherical Peano continuum X is equal to the homotopy dimension of X. We prove the conjecture for one-dimensional or planar Peano continua. This answers a question posed by Cannon and Conner in 2007 concerning the homotopy dimension of planar sets.

In this paper we develop a new approach to the study of uncountable fundamental groups by using Hurewicz fibrations with the unique path-lifting property (lifting spaces for short) as a replacement for covering spaces. In particular, we consider the inverse limit of a sequence of covering spaces of X. It is known that the path-connectivity of the inverse limit can be expressed by means of the derived inverse limit functor (lim) under left arrow (1), which is, however, notoriously difficult to compute when the fundamental group, pi(1)(X), is uncountable. To circumvent this difficulty, we express the set of path-components of the inverse limit (X) over tilde of a sequence of covering spaces in terms of the functors (lim) under left arrow and (lim) under left arrow (1) applied to sequences of countable groups arising from polyhedral approximations of X. A consequence of our computation is that path-connectedness of a lifting space, (X) over tilde, implies that pi(1)(X) supplements pi(1)(X) in pi(1)(X) where pi(1)(X) is the inverse limit of fundamental groups of polyhedral approximations of X. As an application we show that G.Ker(Z)((F) over cap) = (F) over cap not equal G.Ker(B(1,n)) ((F) over cap), where (F) over cap is the canonical inverse limit of finite rank free groups, G is the fundamental group of the Hawaiian Earring, B(1, n) is the Baumslag-Solitar group, and Ker(A)((F) over cap) is the intersection of kernels of homomorphisms from (F) over cap to A. (C) 2021 Elsevier B.V. All rights reserved.

We study the relation between two uncountable groups with remarkable properties (cf. [15]): the topological free product of infinite cyclic groups G (the fundamental group of the Hawaiian Earring), and the inverse limit of finitely generated free groups (F) over capF. The former has a canonical embedding as a proper subgroup of the latter and we examine when G, together with certain naturally defined normal subgroups of (F) over cap generate the entire group (F) over cap. We are interested in particular in normal sub-groups Ker(T) ((F) over cap) = boolean AND{Ker phi vertical bar phi is an element of hom((F) over cap, T)}, where T is some finitely-presented n-slender group. Our main results state that if Tis the infinite cyclic group or the free nilpotent class 2 group on 2 generators, then G and Ker(T)((F) over cap) generate (F) over cap. On the other hand, if T is the free nilpotent class 3 group or a Baumslag-Solitar group, then the product of subgroups G . Ker(T) (F) over cap is a proper subgroup of (F) over cap. In the last section, we provide an interesting geometric interpretation of the above results in terms of path-connectedness of certain fibrations arising as inverse limits of covering spaces over the Hawaiian earring space. (C) 2021 Elsevier Inc. All rights reserved.

In his classical textbook on algebraic topology Edwin Spanier developed the theory of covering spaces within a more general framework of lifting spaces (i.e., Hurewicz fibrations with unique path-lifting property). Among other, Spanier proved that for every space $X$ there exists a universal lifting space, which however need not be simply connected, unless the base space $X$ is semi-locally simply connected. The question on what exactly is the fundamental group of the universal space was left unanswered. The main source of lifting spaces are inverse limits of covering spaces over $X$, or more generally, over some inverse system of spaces converging to $X$. Every metric space $X$ can be obtained as a limit of an inverse system of polyhedra, and so inverse limits of covering spaces over the system yield lifting spaces over $X$. They are related to the geometry (in particular the fundamental group) of $X$ in a similar way as the covering spaces over polyhedra are related to the fundamental group of their base. Thus lifting spaces appear as a natural replacement for the concept of covering spaces over base spaces with bad local properties. In this paper we develop a general theory of lifting spaces and prove that they are preserved by products, inverse limits and other important constructions. We show that maps from $X$ to polyhedra give rise to coverings over $X$ and use that to prove that for a connected, locally path connected and paracompact $X$, the fundamental group of the above-mentioned Spanier's universal space is precisely the intersection of all Spanier groups associated to open covers of $X$, and that the later coincides with the shape kernel of $X$. Furthermore, we examine in more detail lifting spaces over $X$ that arise as inverse limits of coverings over some approximations of $X$.

We study a natural generalization of inverse systems of finite regular covering spaces. A limit of such a system is a fibration whose fibres are profinite topological groups. However, as shown in Conner et al. (Topol Appl 239:234–243, 2018), there are many fibrations whose fibres are profinite groups, which are far from being inverse limits of coverings. We characterize profinite fibrations among a large class of fibrations and relate the profinite topology on the fundamental group of the base with the action of the fundamental group on the fibre, and develop a version of the Borel construction for fibrations whose fibres are profinite groups.

In his classical textbook on algebraic topology Edwin Spanier developed the theory of covering spaces within a more general framework of lifting spaces (i.e., Hurewicz fibrations with unique path-lifting property). Among other, Spanier proved that for every space X there exists a universal lifting space, which however need not be simply connected, unless the base space X is semi-locally simply connected. The question on what exactly is the fundamental group of the universal space was left unanswered. The main source of lifting spaces are inverse limits of covering spaces over X, or more generally, over some inverse system of spaces converging to X. Every metric space X can be obtained as a limit of an inverse system of polyhedra, and so inverse limits of covering spaces over the system yield lifting spaces over X. They are related to the geometry (in particular the fundamental group) of X in a similar way as the covering spaces over polyhedra are related to the fundamental group of their base. Thus lifting spaces appear as a natural replacement for the concept of covering spaces over base spaces with bad local properties. In this paper we develop a general theory of lifting spaces and prove that they are preserved by products, inverse limits and other important constructions. We show that maps from X to polyhedra give rise to coverings over X and use that to prove that for a connected, locally path connected and paracompact X, the fundamental group of the above-mentioned Spanier’s universal space is precisely the intersection of all Spanier groups associated to open covers of X, and that the later coincides with the shape kernel of X. We examine in more detail lifting spaces over X that arise as inverse limits of coverings over some approximations of X. We construct an exact sequence relating the fundamental group of X with the fundamental group and the set of path-components of the lifting space, study relation between the group of deck transformations of the lifting space with the fundamental group of X and prove an existence theorem for lifts of maps into inverse limits of covering spaces. In the final section we consider lifting spaces over non-locally path connected base and relate them to the fibration properties of the so called hat space (or ’Peanification’) construction.

We present new results regarding automatic continuity, unifying some diagonalization concepts that have been developed over the years. For example, any homomorphism from a completely metrizable topological group to Thompson's group F has open kernel. A similar claim holds when F is replaced with a Baumslag-Solitar group or a torsion-free word hyperbolic group.

We show that every homomorphism from the fundamental group of a one-dimensional Peano continuum to the fundamental group of a planar Peano continuum is induced by a continuous map up to conjugation. The set of points at which a planar Peano continuum X is not locally simply connected, which we denote by B(X), is determined solely by the algebraic structure of its fundamental group. Furthermore, we demonstrate how to reconstruct the topological structure of B(X) using only the subgroup lattice of its fundamental group.

An inverse limit of a sequence of covering spaces over a given space $X$ is not, in general, a covering space over $X$ but is still a lifting space, i.e. a Hurewicz fibration with unique path lifting property. Of particular interest are inverse limits of finite coverings (resp. finite regular coverings), which yield fibrations whose fiber is homeomorphic to the Cantor set (resp. profinite topological group). To illustrate the breadth of the theory, we present in this note some curious examples of lifting spaces that cannot be obtained as inverse limits of covering spaces.

The commutator subgroup of a free group is the subgroup consisting of elements contained in the kernel of every homomorphism from the free group to the integers. We give a similar characterization of the second derived subgroup of a free group. Specifically, we show that the second derived subgroup of a free group is equal to the intersection of the kernels of all homomorphisms into a solvable deficiency 1 group.

We will show that every planar Peano continuum whose fundamental group is isomorphic to the fundamental group of a one-dimensional Peano continuum is homotopy equivalent to a one-dimensional Peano continuum. This answers a question asked by Cannon and Conner and illustrates the rigidity of the fundamental group for planar continua.

We show that the first homology group of a locally connected compact metric space is either uncountable or is finitely generated. This is related to Shelah's well-known result which shows that the fundamental group of such a space satisfies a similar criterion. We give an example of such a space whose fundamental group is uncountable but whose first homology is trivial, showing that our result doesn't follow from Shelah's. We clarify a claim made by Pawlikowski and offer a proof of the clarification.

topological spaces have been studied for over a century. A standard tool for studying path connected spaces is the fundamental group. There are questions about fundamental groups of Polish (separable, completely metrizable) spaces that can be answered using descriptive set theory. If one further restricts attention to the fundamental group of a Peano continuum (a connected, locally connected compact metrizable space) then there are interesting dichotomies. In [Sh] Shelah demonstrated the following dichotomy: The fundamental group of a Peano continuum is either finitely generated or of cardinality continuum. Using a theorem by Cannon and Conner [CC] one can replace the word “generated” in the conclusion of Shelah’s theorem with the word “presented”. Pawlikowski later gave a simplified proof of Shelah’s result [P]. Using similar methods Conner and the author obtain the same result for first homology [CoCo]. In this thesis we are going to present improvements on these techniques and numerous applications, which give a great deal of information about fundamental groups of path connected Polish spaces. In Chapter 2 we provide some preliminary definitions and key ideas, as well as some motivation for the results in the succeeding sections. Chapter 3 provides some useful general theorems which provide dichotomies on quotients of the fundamental group. In Chapter 4 we give some examples of topologically defined subgroups of the fundamental group and exhibit upper bounds on their topological complexity. An alternative characterization of the shape kernel is provided for locally path connected spaces. In Chapter 5 we define comonster groups and show an interesting dichotomy on the fundamental group of a Peano continuum. In Chapter 6 we show that subgroups of the fundamental group of a covering space are not more complex than their images under the covering map. Chapter 7 provides