arXiv subjects

Jeremy Brazas

Publications and source records attributed to Jeremy Brazas.

At least 19 recordsLinked to original sources

arXiv2026

Transfinitely iterated wild sets

In this paper, we study homotopical analogues of the Cantor-Bendixson derivative. For each $n\geq 0$, the "$\pi_n$-wild set" $\mathbf{w}_n(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists a shrinking sequence of essential based maps $S^n\to X$. Since the operator $\mathbf{w}_n$ permits iteration, every given space $X$ yields a descending transfinite sequence of nested subspaces $\{\mathbf{w}_n^{\kappa}(X)\}_{\kappa}$ that stabilizes at some smallest ordinal $\mathbf{wrk}_n(X)$ called the "$\pi_n$-wild rank" of $X$. We show that the entire transfinite sequence $\{ho(\mathbf{w}_n^{\kappa}(X))\}_{\kappa}$ of homotopy types is a homotopy invariant of $X$ and that $\mathbf{wrk}_n(X)$ can be an arbitrary countable ordinal when $X$ is an $n$-dimensional Peano continuum. It remains open if there exists a continuum $X$ with uncountable $\pi_n$-wild rank. This difficulty motivates the parallel study a basepoint-free version $\mathbf{fwrk}_n(X)$, called the "free $\pi_n$-wild rank" of $X$. We show that for every continuum $X$, $\mathbf{fwrk}_n(X)$ is always countable and can be any countable ordinal.

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arXiv2026

Tree-like is not a transitive relation on paths

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.

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arXiv2026

Fundamental Groups of Disjointly Tree-Graded Spaces

Tree-graded spaces are a generalization of $\mathbb{R}$-trees and play an important role in describing the large-scale geometry of relatively hyperbolic groups. We consider a subclass of tree-graded spaces that we call "disjointly tree-graded spaces," determined by maps to $\mathbb{R}$-trees. We characterize the fundamental group of a disjointly tree-graded space $(X,\mathscr{P})$ in terms of the fundamental groups of its pieces. Our results apply even in cases where neither $X$ nor its pieces are locally simply connected. In particular, we show that if the pieces are uniformly $1$-$UV_0$, then the fundamental group of a disjointly tree-graded space embeds into the inverse limit of the free products of the fundamental groups of finitely many pieces.

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arXiv2025

Motion Planning on One-Dimensional Peano Continua

We study the Lusternik-Schnirelmann category and topological complexity of 1-dimensional spaces. We define both invariants as lengths of suitable closed filtrations, as opposed to a more common definition based on open covers. Our main results provide a precise description of $\mathbf{cat}(X)$ and $\mathbf{TC}(X)$ for certain 1-dimensional Peano continua $X$ in terms of the wildness rank of $X$. A surprising consequence is that $\mathbf{cat}(X)$ and $\mathbf{TC}(X)$ of a general 1-dimensional space $X$ can be arbitrarily high, which is in stark contrast with the analogous results for 1-dimensional CW-complexes.

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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.

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arXiv2025

Higher homotopy wild sets

The $\pi_n$-wild set $\mathbf{w}_{n}(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists a shrinking sequence of essential based maps $S^n\to X$. In this paper, we show that the homotopy type of $\mathbf{w}_{n}(X)$ is a homotopy invariant of $X$ and, in analogy to the known one-dimensional case, we show that for certain $n$-dimensional $\pi_n$-shape injective metric spaces, the homeomorphism type of $\mathbf{w}_{n}(X)$ is a homotopy invariant of $X$. We also prove that the $\pi_n$-wild set of a Peano continuum can be homeomorphic to any compact metric space.

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arXiv2025

The \v{C}ech homotopy groups of a shrinking wedge of spheres

We compute the \v{C}ech homotopy groups of the $m$-dimensional infinite earring space $\mathbb{E}_m$, i.e. a shrinking wedge of $m$-spheres. In particular, for all $n,m\geq 2$, we prove that $\check{\pi}_n(\mathbb{E}_m)$ is isomorphic to a direct sum of countable powers of homotopy groups of spheres: $\bigoplus_{1\leq j\leq \frac{n-1}{m-1}}\left(\pi_{n}(S^{mj-j+1})\right)^{\mathbb{N}}$. Equipped with this isomorphism and infinite-sum algebra, we also construct new elements of $\pi_n(\mathbb{E}_m)$ with a view toward characterizing the image of the canonical homomorphism $\Psi_{n}:\pi_n(\mathbb{E}_m)\to \check{\pi}_{n}(\mathbb{E}_m)$. We prove that $\Psi_{n}$ is a split epimorphism when $n\leq 2m-1$ and we identify a candidate for the image of $\Psi_n$ when $n>2m-1$.

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arXiv2024

Identities for Whitehead products and infinite sums

Whitehead products and natural infinite sums are prominent in the higher homotopy groups of the $n$-dimensional infinite earring space $\mathbb{E}_n$ and other locally complicated Peano continua. In this paper, we derive general identities for how these operations interact with each other. As an application, we consider a shrinking wedge $X$ of finite $(n-1)$-connected CW-complexes $X_1,X_2,X_3,\dots$ and compute the infinite-sum closure $\mathcal{W}_{2n-1}(X)$ of the set of Whitehead products $[\alpha,\beta]$ in $\pi_{2n-1}\left(X\right)$ where $\alpha,\beta\in\pi_n(X)$ are represented in respective sub-wedges that meet only at the basepoint. In particular, we show that $\mathcal{W}_{2n-1}(X)$ is canonically isomorphic to $\prod_{j=1}^{\infty}\left(\pi_{n}(X_j)\otimes \prod_{k>j}\pi_n(X_k)\right)$. The insight provided by this computation motivates a conjecture about the isomorphism type of the elusive groups $\pi_{2n-1}(\mathbb{E}_n)$, $n\geq 2$.

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arXiv2024

On R-trees, homotopies, and covering maps

A map $p:E\to X$ has the \emph{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 $\mathbb R$-tree $T$ such that the quotient map $p: T\to T/G$ 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.

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arXiv2022

A natural pseudometric on homotopy groups of metric spaces

For a path-connected metric space $(X,d)$, the $n$-th homotopy group $\pi_n(X)$ inherits a natural pseudometric from the $n$-th iterated loop space with the uniform metric. This pseudometric gives $\pi_n(X)$ the structure of a topological group and when $X$ is compact, the induced pseudometric topology is independent of the metric $d$. In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on $\pi_n(X)$. Our main result is that the pseudometric topology agrees with the shape topology on $\pi_n(X)$ if $X$ is compact and $LC^{n-1}$ or if $X$ is an inverse limit of finite polyhedra with retraction bonding maps.

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arXiv2022

Fundamental groups of reduced suspensions are locally free

In this paper, we analyze the fundamental group $\pi_1(\Sigma X,\overline{x_0})$ of the reduced suspension $\Sigma X$ where $(X,x_0)$ is an arbitrary based Hausdorff space. We show that $\pi_1(\Sigma X,\overline{x_0})$ is canonically isomorphic to a direct limit $\varinjlim_{A\in\mathscr{P}}\pi_1(\Sigma A,\overline{x_0})$ where each group $\pi_1(\Sigma A,\overline{x_0})$ is isomorphic to a finitely generated free group or the infinite earring group. A direct consequence of this characterization is that $\pi_1(\Sigma X,\overline{x_0})$ is locally free for any Hausdorff space $X$. Additionally, we show that $\Sigma X$ is simply connected if and only if $X$ is sequentially $0$-connected at $x_0$.

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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 $\mathbb{T}$.

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arXiv2022

Elements of higher homotopy groups undetectable by polyhedral approximation

When non-trivial local structures are present in a topological space $X$, a common approach to characterizing the isomorphism type of the $n$-th homotopy group $\pi_n(X,x_0)$ is to consider the image of $\pi_n(X,x_0)$ in the $n$-th \v{C}ech homotopy group $\check{\pi}_n(X,x_0)$ under the canonical homomorphism $\Psi_{n}:\pi_n(X,x_0)\to \check{\pi}_n(X,x_0)$. The subgroup $\ker(\Psi_n)$ is the obstruction to this tactic as it consists of precisely those elements of $\pi_n(X,x_0)$, which cannot be detected by polyhedral approximations to $X$. In this paper, we use higher dimensional analogues of Spanier groups to characterize $\ker(\Psi_n)$. In particular, we prove that if $X$ is paracompact, Hausdorff, and $LC^{n-1}$, then $\ker(\Psi_n)$ is equal to the $n$-th Spanier group of $X$. We also use the perspective of higher Spanier groups to generalize a theorem of Kozlowski-Segal, which gives conditions ensuring that $\Psi_{n}$ is an isomorphism.

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arXiv2022

Constructing arcs from paths using Zorn's Lemma

It is a well-known fact that every path-connected Hausdorff space is arcwise connected. Typically, this result is viewed as a consequence of a sequence of fairly technical results from continuum theory. In this note, we exhibit a direct and simple proof of this statement, which makes explicit use of Zorn's Lemma. Additionally, by carefully breaking down the proof, we identify a modest improvement to a class of spaces relevant to algebraic topology.

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arXiv2022

Homotopy groups of shrinking wedges of non-simply connected CW-complexes

In this paper, we study the homotopy groups of a shrinking wedge $X$ of a sequence $\{X_j\}$ of non-simply connected CW-complexes. Using a combination of generalized covering space theory and shape theory, we construct a canonical homomorphism $$\Theta:\pi_n(X)\to\prod_{j\in\mathbb{N}}\bigoplus_{\pi_1(X)/\pi_1(X_j)}\pi_n(X_j),$$ characterize its image, and prove that $\Theta$ is injective whenever each universal cover $\widetilde{X}_j$ is $(n-1)$-connected. These results (1) provide a characterization of the $n$-th homotopy group of the shrinking wedge of copies of $\mathbb{RP}^n$, (2) provide a characterization of $\pi_2$ of an arbitrary shrinking wedge, and (3) imply that a shrinking wedge of aspherical CW-complexes is aspherical.

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arXiv2021

Free Quasitopological Groups

In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group $F_q(X)$ on a space $X$. We show that free quasitopological groups may be constructed directly as quotient spaces of free semitopological monoids, which are themselves constructed by iterating product spaces equipped with the "cross topology." Using this explicit description of $F_q(X)$, we show that for any $T_1$ space $X$, $F_q(X)$ is the direct limit of closed subspaces $F_q(X)_n$ of words of length at most $n$. We also prove that the natural map ${\bf i_n}:\coprod_{i=0}^{n}(X\sqcup X^{-1})^{\otimes i}\to F_q(X)_n$ is quotient for all $n\geq 0$. Equipped with this convenient characterization of the topology of free quasitopological groups, we show, among other things, that a subspace $Y\subseteq X$ is closed if and only if the inclusion $Y\to X$ induces a closed embedding $F_q(Y)\to F_q(X)$ of free quasitopological groups.

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arXiv2021

Sequential $n$-connectedness and infinite factorization in higher homotopy groups

A space $X$ is "sequentially $n$-connected" at $x\in X$ if for every $0\leq k\leq n$ and sequence of maps $f_1,f_2,f_3,\dots:S^k\to X$ that converges toward a point $x\in X$, the maps $f_m$ contract by a sequence of null-homotopies that converge toward $x$. We use this property, in conjunction with the Whitney Covering Lemma, as a foundation for developing new methods for characterizing higher homotopy groups of finite dimensional Peano continua. Among many new computations, a culminating result of this paper is: if $Y$ is a space obtained by attaching an infinite shrinking sequence $A_1,A_2,A_3,\dots$ of $(n-1)$-connected CW-complexes to a one-dimensional Peano continuum $X$ along a sequence of points in $X$, then there is an injection $Φ:π_n(Y)\to \prod_{j=1}^{\infty}\bigoplus_{π_1(X)}π_n(A_j)$ that is canonical after a certain choice of paths in $X$ is made. Moreover, we characterize the image of $Φ$ using generalized covering space theory. As a case of particular interest, this provides a characterization of $π_n(\mathbb{H}_1\vee \mathbb{H}_n)$ where $\mathbb{H}_n$ denotes the $n$-dimensional Hawaiian earring.

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arXiv2020

The infinitary n-cube shuffle

In this paper, we formalize the sense in which higher homotopy groups are "infinitely commutative." In particular, we both simplify and extend the highly technical procedure, due to Eda and Kawamura, for constructing homotopies that isotopically rearrange infinite configurations of disjoint $n$-cubes within the unit $n$-cube.

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