Searcharxiv⌕ Search

arXiv subjects

Hanspeter Fischer

Publications and source records attributed to Hanspeter Fischer.

11 recordsLinked to original sources
arXiv2025

Nonlocal loss of first homotopy in polyhedral approximations of Peano continua

If a Peano continuum $X$ is semilocally simply connected, then it has a finite polyhedral approximation whose fundamental group is isomorphic to that of $X$. In general, this fails to be true. It is known that the fundamental group of a locally complicated Peano continuum may contain nontrivial elements that are persistently undetectable by polyhedral approximations, at all scales. However, we show that such failure is not inherently local.

math.AT↗
arXiv2022

A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space

We present a 2-dimensional Peano continuum $\mathbb{T}\subseteq \mathbb{R}^3$ with the following properties: (1) There is a universal covering projection $q:\overline{\mathbb{T}}\rightarrow \mathbb{T}$ with uncountable fundamental group $\pi_1(\overline{\mathbb{T}})$; (2) For every $1\not=[\overline{\alpha}]\in \pi_1(\overline{\mathbb{T}},\ast)$, there is a covering projection $r:(E,e)\rightarrow (\overline{\mathbb{T}},\ast)$ such that $[\overline{\alpha}]\not\in r_\#\pi_1(E,e)$; (3) There is no universal covering projection $r:E\rightarrow \overline{\mathbb{T}}$; (4) The universal object $p:\widetilde{\mathbb{T}}\rightarrow \mathbb{T}$ in the category of fibrations with unique path lifting (and path-connected total space) over $\mathbb{T}$ has trivial fundamental group $\pi_1(\widetilde{\mathbb{T}})=1$; (5) $p:\widetilde{\mathbb{T}}\rightarrow \mathbb{T}$ is not a path component of an inverse limit of covering projections over $\mathbb{T}$.

math.AT↗
arXiv2020

On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements

We construct a space $\mathbb{P}$ for which the canonical homomorphism $π_1(\mathbb{P},p) \rightarrow \checkπ_1(\mathbb{P},p)$ from the fundamental group to the first Čech homotopy group is not injective, although it has all of the following properties: (1) $\mathbb{P}\setminus\{p\}$ is a 2-manifold with connected non-compact boundary; (2) $\mathbb{P}$ is connected and locally path connected; (3) $\mathbb{P}$ is strongly homotopically Hausdorff; (4) $\mathbb{P}$ is homotopically path Hausdorff; (5) $\mathbb{P}$ is 1-UV$_0$; (6) $\mathbb{P}$ admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) $π_1(\mathbb{P},p)$ naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring; (8) $π_1(\mathbb{P},p)$ is locally free.

math.AT↗
arXiv2019

Fibrations, unique path lifting, and continuous monodromy

Given a path-connected space $X$ and $H\leqπ_1(X,x_0)$, there is essentially only one construction of a map $p_H:(\widetilde{X}_H,\widetilde{x}_0)\rightarrow(X,x_0)$ with connected and locally path-connected domain that can possibly have the following two properties: $(p_{H})_{\#}π_1(\widetilde{X}_H,\widetilde{x}_0)=H$ and $p_H$ has the unique lifting property. $\widetilde{X}_H$ consists of equivalence classes of paths starting at $x_0$, appropriately topologized, and $p_H$ is the endpoint projection. For $p_H$ to have these two properties, $T_1$ fibers are necessary and unique path lifting is sufficient. However, $p_H$ always admits the standard lifts of paths. We show that $p_H$ has unique path lifting if it has continuous (standard) monodromies toward a $T_1$ fiber over $x_0$. Assuming, in addition, that $H$ is locally quasinormal (e.g., if $H$ is normal) we show that $X$ is homotopically path Hausdorff relative to $H$. We show that $p_H$ is a fibration if $X$ is locally path connected, $H$ is locally quasinormal, and all (standard) monodromies are continuous.

math.AT↗
arXiv2018

Test map characterizations of local properties of fundamental groups

Local properties of the fundamental group of a path-connected topological space can pose obstructions to the applicability of covering space theory. A generalized covering map is a generalization of the classical notion of covering map defined in terms of unique lifting properties. The existence of generalized covering maps depends entirely on the verification of the unique path lifting property for a standard covering construction. Given any path-connected metric space $X$, and a subgroup $H\leqπ_1(X,x_0)$, we characterize the unique path lifting property relative to $H$ in terms of a new closure operator on the $π_1$-subgroup lattice that is induced by maps from a fixed "test" domain into $X$. Using this test map framework, we develop a unified approach to comparing the existence of generalized coverings with a number of related properties.

math.AT↗
arXiv2014

Cotorsion-free groups from a topological viewpoint

We present a characterization of cotorsion-free abelian groups in terms of homomorphisms from fundamental groups of Peano continua, which aligns naturally with the generalization of slenderness to non-abelian groups. In the process, we calculate the first homology group of the Griffiths twin cone.

math.AT↗
arXiv2013

Word calculus in the fundamental group of the Menger curve

The fundamental group of the Menger universal curve is uncountable and not free, although all of its finitely generated subgroups are free. It contains an isomorphic copy of the fundamental group of every one-dimensional separable metric space and an isomorphic copy of the fundamental group of every planar Peano continuum. We give an explicit and systematic combinatorial description of the fundamental group of the Menger universal curve and its generalized Cayley graph in terms of word sequences. The word calculus, which requires only two letters and their inverses, is based on Pasynkov's partial topological product representation and can be expressed in terms of a variation on the classical puzzle known as the Towers of Hanoi.

math.GT↗
arXiv2013

A core-free semicovering of the Hawaiian Earring

The connected covering spaces of a connected and locally path-connected topological space $X$ can be classified by the conjugacy classes of those subgroups of $π_1(X,x)$ which contain an open normal subgroup of $π_1(X,x)$, when endowed with the natural quotient topology of the compact-open topology on based loops. There are known examples of semicoverings (in the sense of Brazas) that correspond to open subgroups which do not contain an open normal subgroup. We present an example of a semicovering of the Hawaiian Earring $\mathds{H}$ with corresponding open subgroup of $π_1(\mathds{H})$ which does not contain {\em any} nontrivial normal subgroup of $π_1(\mathds{H})$.

math.GT↗
arXiv2011

On semilocally simply connected spaces

The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via the modified Spanier group semilocal simple connectivity can be characterized; and (iii) to point out that with just a slightly modified definition of semilocal simple connectivity which is sometimes also used in literature, the classical Spanier group gives the correct characterization within the general class of path-connected topological spaces. While the condition "semilocally simply connected" plays a crucial role in classical covering theory, in generalized covering theory one needs to consider the condition "homotopically Hausdorff" instead. The paper also discusses which implications hold between all of the abovementioned conditions and, via the modified Spanier groups, it also unveils the weakest so far known algebraic characterization for the existence of generalized covering spaces as introduced by Fischer and Zastrow. For most of the implications, the paper also proves the non-reversibility by providing the corresponding examples. Some of them rely on spaces that are newly constructed in this paper.

math.GT↗
arXiv2005

The fundamental groups of subsets of closed surfaces inject into their first shape groups

We show that for every subset X of a closed surface M^2 and every basepoint x_0, the natural homomorphism from the fundamental group to the first shape homotopy group, is injective. In particular, if X is a proper compact subset of M^2, then pi_1(X,x_0) is isomorphic to a subgroup of the limit of an inverse sequence of finitely generated free groups; it is therefore locally free, fully residually free and residually finite.

math.GR↗