44 citations · 129 across the 32 of their papers we have counts for
34 papers
Transfinitely iterated wild sets
Jeremy Brazas, Atish Mitra
In this paper, we study homotopical analogues of the Cantor-Bendixson derivative. For each , the "-wild set" of a topological space is the subsp…
Tree-like is not a transitive relation on paths
Jeremy Brazas, Gregory R. Conner, Paul Fabel +1
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 p…
Motion Planning on One-Dimensional Peano Continua
Jeremy Brazas, Petar Pavesic
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…
Nonlocal loss of first homotopy in polyhedral approximations of Peano continua
Jeremy Brazas, Hanspeter Fischer
If a Peano continuum is semilocally simply connected, then it has a finite polyhedral approximation whose fundamental group is isomorphic to that of . In general, this fails…
Higher homotopy wild sets
Jeremy Brazas, Atish Mitra
The -wild set of a topological space is the subspace of consisting of the points at which there exists a shrinking sequence of essential based maps…
The Čech homotopy groups of a shrinking wedge of spheres
Jeremy Brazas
We compute the Čech homotopy groups of the -dimensional infinite earring space , i.e. a shrinking wedge of -spheres. In particular, for all , we prov…
Identities for Whitehead products and infinite sums
Jeremy Brazas
Whitehead products and natural infinite sums are prominent in the higher homotopy groups of the -dimensional infinite earring space and other locally complicated…
On R-trees, homotopies, and covering maps
Jeremy Brazas, Gregory R. Conner, Paul Fabel +1
A map has the \emph{unique path lifting} property if every path in , after a choice of an initial point, lifts uniquely to a path in . We prove that if a group …
A natural pseudometric on homotopy groups of metric spaces
Jeremy Brazas, Paul Fabel
For a path-connected metric space , the -th homotopy group inherits a natural pseudometric from the -th iterated loop space with the uniform metric. This pseu…
Fundamental groups of reduced suspensions are locally free
Jeremy Brazas, Patrick Gillespie
In this paper, we analyze the fundamental group of the reduced suspension where is an arbitrary based Hausdorff space. We show that $π_1(ΣX,…
A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space
Jeremy Brazas, Hanspeter Fischer
We present a 2-dimensional Peano continuum with the following properties: (1) There is a universal covering projection $q:\overline{\mathbb{T}}\r…
Elements of higher homotopy groups undetectable by polyhedral approximation
John K. Aceti, Jeremy Brazas
When non-trivial local structures are present in a topological space , a common approach to characterizing the isomorphism type of the -th homotopy group is to c…
Homotopy groups of shrinking wedges of non-simply connected CW-complexes
Jeremy Brazas
In this paper, we study the homotopy groups of a shrinking wedge of a sequence of non-simply connected CW-complexes. Using a combination of generalized covering space…
Free Quasitopological Groups
Jeremy Brazas, Sarah Emery
In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group on a space . We show t…
Sequential -connectedness and infinite factorization in higher homotopy groups
Jeremy Brazas
A space is "sequentially -connected" at if for every and sequence of maps that converges toward a point , the m…
The infinitary n-cube shuffle
Jeremy Brazas
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 t…
On maps with continuous path lifting
Jeremy Brazas, Atish Mitra
We study a natural generalization of covering projections defined in terms of unique lifting properties. A map has the "continuous path-covering property" if all paths i…
Infinitary commutativity and fundamental groups of topological monoids
Jeremy Brazas, Patrick Gillespie
The well-known Eckmann-Hilton Principle may be applied to prove that fundamental groups of -spaces are commutative. In this paper, we identify an infinitary analogue of the Eckm…
Transfinite product reduction in fundamental groupoids
Jeremy Brazas
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…
Dense products in fundamental groupoids
Jeremy Brazas
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 u…
Scattered products in fundamental groupoids
Jeremy Brazas
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-defi…
On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements
Jeremy Brazas, Hanspeter Fischer
We construct a space for which the canonical homomorphism from the fundamental group to the first Čech homotopy…
On the discontinuity of the -action
Jeremy Brazas
We show the classical -action on the -th homotopy group can fail to be continuous for any when the homotopy groups are equipped with the natural quotient topology. In p…
Realizing spaces as path-component spaces
Taras Banakh, Jeremy Brazas
The path component space of a topological space is the quotient space whose points are the path components of . We show that every Tychonoff space is the path-c…