arXiv subjects

Jeremy Brazas

Publications and source records attributed to Jeremy Brazas.

36 records · Page 2Linked to original sources

arXiv2020

On maps with continuous path lifting

We study a natural generalization of covering projections defined in terms of unique lifting properties. A map $p:E\to X$ has the "continuous path-covering property" if all paths in $X$ lift uniquely and continuously (rel. basepoint) with respect to the compact-open topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological $π_1$: for any path-connected Hausdorff space $X$, maps $E\to X$ with the continuous path-covering property are classified up to weak equivalence by subgroups $H\leq π_1(X,x_0)$ with totally path-disconnected coset space $π_1(X,x_0)/H$. Here, "weak equivalence" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.

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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.

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arXiv2020

Transfinite product reduction in fundamental groupoids

Infinite products, indexed by countably infinite linear orders, arise naturally in the context of fundamental groupoids. Such products are called "transfinite" if the index orders are permitted to contain a dense suborder and are called "scattered" otherwise. In this paper, we prove several technical lemmas related to the reduction (i.e. combining of factors) of transfinite products in fundamental groupoids. Applying these results, we show that if the transfinite fundamental group operations are well-defined in a space $X$ with a scattered algebraic $1$-wild set $\mathbf{aw}(X)$, then all transfinite fundamental groupoid operations are also well-defined.

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arXiv2020

Infinitary commutativity and fundamental groups of topological monoids

The well-known Eckmann-Hilton Principle may be applied to prove that fundamental groups of $H$-spaces are commutative. In this paper, we identify an infinitary analogue of the Eckmann-Hilton Principle that applies to fundamental groups of all topological monoids and slightly more general objects called pre-$Δ$-monoids. In particular, we show that every pre-$Δ$-monoid $M$ is "transfinitely $π_1$-commutative" in the sense that permutation of the factors of any infinite loop-concatenation indexed by a countably infinite order and based at the identity $e\in M$ is a homotopy invariant action. We also give a detailed account of fundamental groups of James reduced products and apply transfinite $π_1$-commutativity to make several computations.

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arXiv2019

Dense products in fundamental groupoids

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally path-connected metric space, the well-definedness of countable dense products in the fundamental group need not imply the well-definedness of countable dense products in the fundamental groupoid. Additionally, we show the fundamental groupoid $Π_1(X)$ has well-defined dense products if and only if $X$ admits a generalized universal covering space.

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arXiv2019

Scattered products in fundamental groupoids

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. We prove that the well-definedness of products indexed by a scattered linear order in the fundamental groupoid of a first countable space is equivalent to the homotopically Hausdorff property. To prove this characterization, we employ the machinery of closure operators, on the $π_1$-subgroup lattice, defined in terms of test maps from one-dimensional domains.

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arXiv2018

On the discontinuity of the $π_1$-action

We show the classical $π_1$-action on the $n$-th homotopy group can fail to be continuous for any $n$ when the homotopy groups are equipped with the natural quotient topology. In particular, we prove the action $π_1(X)\timesπ_n(X)\toπ_n(X)$ fails to be continuous for a one-point union $X=A\vee \mathbb{H}_n$ where $A$ is an aspherical space such that $π_1(A)$ is a topological group and $\mathbb{H}_n$ is the $(n-1)$-connected, n-dimensional Hawaiian earring space $\mathbb{H}_n$ for which $π_n(\mathbb{H}_n)$ is a topological abelian group.

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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.

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arXiv2018

Realizing spaces as path-component spaces

The path component space of a topological space $X$ is the quotient space $π_0(X)$ whose points are the path components of $X$. We show that every Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of weight $w(Y)=w(X)$ such that the natural quotient map $Y\to π_0(Y)=X$ is a perfect map. Hence, many topological properties of $X$ transfer to $Y$. We apply this result to construct a compact space $X\subset \mathbb{R}^3$ for which the fundamental group $π_1(X,x_0)$ is an uncountable, cosmic, $k_ω$-topological group but for which the canonical homomorphism $ψ:π_1(X,x_0)\to \checkπ_1(X,x_0)$ to the first shape homotopy group is trivial.

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arXiv2015

Generalized covering space theories

In this paper, we unify various approaches to generalized covering space theory by introducing a categorical framework in which coverings are defined purely in terms of unique lifting properties. For each category $\mathcal{C}$ of path-connected spaces having the unit disk as an object, we construct a category of $\mathcal{C}$-coverings over a given space $X$ that embeds in the category of $π_1(X,x_0)$-sets via the usual monodromy action on fibers. When $\mathcal{C}$ is extended to its coreflective hull $\mathscr{H}(\mathcal{C})$, the resulting category of based $\mathscr{H}(\mathcal{C})$-coverings is complete, has an initial object, and often characterizes more of the subgroup lattice of $π_1(X,x_0)$ than traditional covering spaces. We apply our results to three special coreflective subcategories: (1) The category of $Δ$-coverings employs the convenient category of $Δ$-generated spaces and is universal in the sense that it contains every other generalized covering category as a subcategory. (2) In the locally path-connected category, we preserve notion of generalized covering due to Fischer and Zastrow and characterize the topology of such coverings using the standard whisker topology. (3) By employing the coreflective hull $\mathbf{Fan}$ of the category of all contractible spaces, we characterize the notion of continuous lifting of paths and identify the topology of $\mathbf{Fan}$-coverings as the natural quotient topology inherited from the path space.

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arXiv2014

Strongly Pseudoradial Spaces

The "weakly Hausdorff" property for pseudoradial spaces fails to be naturally characterized by unique convergence of transfinite sequences. In response, we develop the category $\mathbf{SPsRad}$ of strongly pseudoradial spaces, compactly generated spaces whose closed sets are determined by globally continuous maps from well-ordered spaces. Categorically, $\mathbf{SPsRad}$ is the coreflective hull of the class of well-ordered spaces, and $\mathbf{SPsRad}$ is Cartesian closed. The strongly pseudoradial weakly Hausdorff spaces admit a natural characterization involving unique extensions of injective maps of well-ordered spaces. We also obtain analogs in $\mathbf{SPsRad}$ of the fact that for sequential spaces, sequential compactness is equivalent to countable compactness.

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arXiv2013

On fundamental groups with the quotient topology

The quasitopological fundamental group $π_{1}^{qtop}(X,x_0)$ is the fundamental group endowed with the natural quotient topology inherited from the space of based loops and is typically non-discrete when $X$ does not admit a traditional universal cover. This topologized fundamental group is an invariant of homotopy type which has the ability to distinguish weakly homotopy equivalent and shape equivalent spaces. In this paper, we clarify various relationships among topological properties of the group $π_{1}^{qtop}(X,x_0)$ and properties of the underlying space $X$ such as `$π_{1}$-shape injectivity' and `homotopically path-Hausdorff.' A space $X$ is $π_1$-shape injective if the fundamental group canonically embeds in the first shape group so that the elements of $π_1(X,x_0)$ can be represented as sequences in an inverse limit. We show a locally path connected metric space $X$ is $π_1$-shape injective if and only if $π_{1}^{qtop}(X,x_0)$ is invariantly separated in the sense that the intersection of all open invariant (i.e. normal) subgroups is the trivial subgroup. In the case that $X$ is not $π_1$-shape injective, the homotopically path-Hausdorff property is useful for distinguishing homotopy classes of loops and guarantees the existence of certain generalized covering maps. We show that a locally path connected space $X$ is homotopically path-Hausdorff if and only if $π_{1}^{qtop}(X,x_0)$ satisfies the $T_1$ separation axiom.

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arXiv2013

Open subgroups of free topological groups

The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We apply the more general theory of semicovering spaces to obtain analogous subgroup theorems for topological groups: Every open subgroup of a free Graev topological group is a free Graev topological group. An open subgroup of a free Markov topological group is a free Markov topological group if and only if it is disconnected.

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arXiv2012

Thick Spanier groups and the first shape group

We develop a new route through which to explore $\kerΨ_X$, the kernel of the $π_1$-shape group homomorphism determined by a general space $X$, and establish, for each locally path connected, paracompact Hausdorff space $X$, $\kerΨ_X$ is precisely the Spanier group of $X$.

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arXiv2012

The fundamental group as a topological group

This paper is devoted to the study of a natural group topology on the fundamental group which remembers local properties of spaces forgotten by covering space theory and weak homotopy type. It is known that viewing the fundamental group as the quotient of the loop space often fails to result in a topological group; we use free topological groups to construct a topology which promotes the fundamental group of any space to topological group structure. The resulting invariant, denoted $π_{1}^τ$, takes values in the category of topological groups, can distinguish spaces with isomorphic fundamental groups, and agrees with the quotient fundamental group precisely when the quotient topology yields a topological group. Most importantly, this choice of topology allows us to naturally realize free topological groups and pushouts of topological groups as fundamental groups via topological analogues of classical results in algebraic topology.

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arXiv2011

Semicoverings: a generalization of covering space theory

The fundamental groupoid of a space becomes enriched over the category of topological spaces when the hom-sets are endowed with topologies intimately related to universal constructions of topological groups. This paper is devoted to a generalization of classical covering theory in the context of this construction.

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arXiv2010

The topological fundamental group and free topological groups

The topological fundamental group $π_{1}^{top}$ is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary space $X$, we compute the topological fundamental group of the suspension space $Σ(X_+)$ and find that $π_{1}^{top}(Σ(X_+))$ either fails to be a topological group or is the free topological group on the path component space of $X$. Using this computation, we provide an abundance of counterexamples to the assertion that all topological fundamental groups are topological groups. A relation to free topological groups allows us to reduce the problem of characterizing Hausdorff spaces $X$ for which $π_{1}^{top}(Σ(X_+))$ is a Hausdorff topological group to some well known classification problems in topology.

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arXiv2010

The Topological Fundamental Group and Hoop Earring Spaces

The topological fundamental group $π_{1}^{top}$ is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased space $X$, we compute the topological fundamental group of the "hoop earring" space of $X$, which is the reduced suspension of $X$ with disjoint basepoint. We do so by factorizing the quotient map $Ω(ΣX_{+},x)\to π_{1}^{top}(ΣX_{+},x)$ through a free topological monoid with involution $M(X)$ such that the map $M(X)\shortrightarrow π_{1}^{top}(ΣX_{+},x)$ is also a quotient map. $π_{1}^{top}(ΣX_{+},x)$ is T1 and an embedding $X\shortrightarrow π_{1}^{top}(ΣX_{+},x)$ illustrates that $π_{1}^{top}(ΣX_{+},x)$ is not a topological group when $X$ is not regular. These hoop earring spaces provide a simple class of counterexamples to the claim that $π_{1}^{top}$ is a functor to the category of topological groups.

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