arXiv subjects

Jesús González

Publications and source records attributed to Jesús González.

At least 19 recordsLinked to original sources

arXiv2026

A combinatorial genesis of the right-angled relations in Artin's classical braid groups

The configuration space $\text{UC}(n,p\times q)$ of $n$ unlabelled non-overlapping unit squares in a $p\times q$ rectangle is known to recover the homotopy type of the classical configuration space of $n$ unlabelled points in the plane, provided $\min\{p,q\}\geq n$. Thus the fundamental group $B_n(p\times q)$ of $\text{UC}(n,p\times q)$ yields a $(p,q)$-approximation of Artin's classical braid group $B_n$. We describe a right-angled Artin group presentation for $B_n(p\times q)$ in cases where $\text{UC}(n,p\times q)$ is known to be aspherical. When $\min\{p,q\}=2$, our presentation agrees with Artin's classical presentation for $B_n$ removing the Artin-Tits relations. This allows us to deduce the value of the Lusternik-Schnirelmann category of the corresponding aspherical spaces $\text{UC}(n,p\times q)$, as well as the values of all their $k$-sequential topological complexities, both in the classical (Rudyak et al.) and distributional (Dransihnikov et al.) contexts.

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arXiv2026

On the sequential topological complexity and the LS-category of the cofiber of higher diagonals for symmetric products of non-orientable surfaces

For positive integers $k$, $n$, and $g$ with $k\geq2$, we give a closed-form expression for the $k$-th $\mathbb{Z}_2$-zero-divisor cup length $\mathsf{zcl}_k(SP^n(N_g))$ of the $n$-th symmetric product $SP^n(N_g)$ of the closed non-orientable surface $N_g$ of genus $g$. This allows us to estimate, and in some cases, completely determine, the $k$-th sequential topological complexity $\mathsf{TC}_k(SP^n(N_g))$, as well as the Lusternik--Schnirelmann category of the homotopy cofiber of the $k$-th diagonal map $SP^n(N_g) \to (SP^n(N_g))^k$. Our results recover previously known facts for even-dimensional real projective spaces ($g=1$) and closed non-orientable surfaces ($n=1$). In addition, we show that, as $g$ grows, $\mathsf{TC}_2(SP^n(N_g))$ behaves in a different way as all other invariants $\mathsf{TC}_k(SP^n(N_g))$ do. Likewise, as $k$ grows, we describe an eventual maximal-possible linear growth of $\mathsf{zcl}_k(SP^n(N_g))$, which allows us to prove the rationality conjecture of Farber and Oprea for the TC-generating function of $SP^n(N_g)$.

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arXiv2026

The 15 Puzzle and homological stability in the space direction

The ordered configuration space of $n$ open unit squares in the $w$ by $h$ rectangle exhibits homological stability in the space direction. That is, for fixed $n$ and fixed homological degree $k$, once the underlying rectangle is large enough, making it any larger does not change the $k$-th homology of the square configuration space. In this paper, we sharpen the stable range. Finding bounds for $w$ and $h$ in terms of $n$ and $k$, we prove that most rectangles can be almost entirely filled with squares and there still be an isomorphism between the $k$-th homology of the resulting square configuration space and the $k$-th homology of the ordered configuration space of $n$ points in the plane.

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arXiv2025

Square-section braid groups and Higman-Neumann-Neumann extensions

For positive integers $n$, $p$ and $q$ with $pq-n>0$, let $UC(n,p\times q)$ denote the configuration space of $n$ unlabelled hard unit squares in the rectangle $[0,p]\times[0,q]$, and let $B_n(p\times q)$ denote the corresponding fundamental group. It is known that, as $pq-n$ becomes large, $UC(n,p\times q)$ starts capturing homotopical properties of the classical configuration space of $n$ unlabelled pairwise-distinct points in the plane. At the start of this approximation process, $UC(pq-1,p\times q)$ is homotopy equivalent to a wedge of $(p-1)(q-1)$ circles, while the only other general families of spaces $UC(n,p\times q)$ known to be aspherical are $UC(n,p\times2)$ for $p\geq n$, and $UC(pq-2,p\times q)$. The fundamental groups of the former family are known to be responsible for the ``right-angled'' relations in Artin's classical braid groups. We prove that the fundamental groups of the latter family have a minimal presentation all whose relators are commutators. In particular, after explaining how $B_{2p-2}(p\times2)$ arises as the right-angled Artin group (RAAG) associated to a certain meta-edge, we show that $B_{3p-2}(p\times3)$ is a Higman-Neumann-Neumann extension of the RAAG associated to the corresponding meta-square. We provide a geometric interpretation of the latter fact in terms of Salvetti complexes.

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arXiv2024

Borsuk--Ulam property for graphs II: The $\mathbb{Z}_n$-action

For a finite group $H$ and connected topological spaces $X$ and $Y$ such that $X$ is endowed with a free left $H$-action $τ$, we provide a geometric condition in terms of the existence of a commutative diagram of spaces (arising from the triple $(X,Y;τ)$) to decide whether the Borsuk--Ulam property holds for based homotopy classes $α\in[X,Y]_0$, as well as for free homotopy classes $α\in[X,Y]$. Here, a homotopy class $α$ is said to satisfy the Borsuk--Ulam property if, for each of its representatives $f\inα$, there exists an $H$-orbit where $f$ fails to be injective. Our geometric characterization is attained by constructing an $H$-equivariant map from $X$ to the classical configuration space $F_{|H|}(Y)$. We derive an algebraic condition from the geometric characterisation, and show that the former one is in fact equivalent to the latter one when $X$ and $Y$ are aspherical. We then specialize to the 1-dimensional case, i.e., when $X$ is an arbitrary connected graph, $H$ is cyclic, and $Y$ is either an interval, a circle, or their wedge sum. The graph-braid-group ingredient in our characterizations is then effectively controlled through the use of discrete Morse theory.

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arXiv2024

Abrams' stabilization theorem for no-k-equal configuration spaces on graphs

For a graph $G$, let Conf$(G,n)$ denote the classical configuration space of $n$ labelled points in $G$. Abrams introduced a cubical complex, denoted here by DConf$(G,n)$, sitting inside Conf$(G,n)$ as a strong deformation retract provided $G$ is suitably subdivided. Using discrete Morse Theory techniques, we extend Abrams' result to the realm of configurations having no $k$-fold collisions.

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arXiv2024

The rational (non-)formality of the non-3-equal manifolds

Let $M^{(k)}_{d}(n)$ be the manifold of $n$-tuples $(x_1,\ldots,x_n)\in(\mathbb{R}^d)^n$ having non-$k$-equal coordinates. We show that, for $d\geq2$, $M^{(3)}_{d}(n)$ is rationally formal if and only if $n\leq 6$. This stands in sharp contrast with the fact that all classical configuration spaces $M^{(2)}_d(n)=\text{Conf}(\hspace{.2mm}\mathbb{R}^d,n)$ are rationally formal, just as are all complements of arrangements of arbitrary complex subspaces with geometric lattice of intersections. The rational non formality of $M^{(3)}_{d}(n)$ for $n>6$ is established via detection of non-trivial triple Massey products assessed through Poincaré duality.

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arXiv2023

Parametrised collision-free optimal motion planning algorithms in Euclidean spaces

We describe parametrised motion planning algorithms for systems controlling objects represented by points that move without collisions in an even dimensional Euclidean space and in the presence of up to three obstacles with \emph{a priori} unknown positions. Our algorithms are optimal in the sense that the parametrised local planners have minimal posible size.

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arXiv2023

Higher topological complexity of a map

The higher topological complexity of a space $X$, $\text{TC}_r(X)$, $r=2,3,\ldots$, and the topological complexity of a map $f$, $\text{TC}(f)$, have been introduced by Rudyak and Pavešić, respectively, as natural extensions of Farber's topological complexity of a space. In this paper we introduce a notion of higher topological complexity of a map~$f$, $\text{TC}_{r,s}(f)$, for $1\leq s\leq r\geq2$, which simultaneously extends Rudyak's and Pavešić's notions. Our unified concept is relevant in the $r$-multitasking motion planning problem associated to a robot devise when the forward kinematics map plays a role in $s$ prescribed stages of the motion task. We study the homotopy invariance and the behavior of $\text{TC}_{r,s}$ under products and compositions of maps, as well as the dependence of $\text{TC}_{r,s}$ on $r$ and $s$. We draw general estimates for $\text{TC}_{r,s}(f\colon X\to Y)$ in terms of categorical invariants associated to $X$, $Y$ and $f$. In particular, we describe within one the value of $\text{TC}_{r,s}$ in the case of the non-trivial double covering over real projective spaces, as well as for their complex counterparts.

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arXiv2023

Secondary School Students observe Venus with NASA Infrared Telescope Facility (IRTF)

Astronomy and astrophysics are regarded as highly motivating topics for students in primary and secondary schools, and they have been a recurrent and effective resource to inspire passion about science. In fact, during the last years we have witnessed a boost of facilities providing small robotic telescopes for teachers and students to remotely undertake their own observing projects. A step forward is presented here, where we describe the experience of secondary school students attending professional observations of Venus at NASA's Infrared Telescope Facility (IRTF) and, in a second observing run, conducting the observations by themselves. In addition to quickly mastering the basic operation of the control software for the SpeX instrument, the students successfully performed different types of data acquisition, including drift scan imaging.

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arXiv2022

Borsuk-Ulam property for graphs

For finite connected graphs $Γ$ and $G$, with $Γ$ admitting a free involution $τ$, we characterize the based homotopy classes $α\in[Γ,G]$ for which the Borsuk-Ulam property holds in the sense of Gonçalves, Guaschi and Casteluber-Laass, i.e., the homotopy classes $α$ so that each of its representatives $f\inα$ satisfies $f(x) = f(τ\cdot x)$ for some $x\inΓ$. This is attained through a graph-braid-group perspective aided by the use of discrete Morse theory.

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arXiv2022

An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs

We introduce an algorithm that constructs a discrete gradient field on any simplicial complex. We show that, in all situations, the gradient field is maximal possible and, in a number of cases, optimal. We make a thorough analysis of the resulting gradient field in the case of Munkres' discrete model for $\text{Conf}(K_m,2)$, the configuration space of ordered pairs of non-colliding particles on the complete graph $K_m$ on $m$ vertices. Together with the use of Forman's discrete Morse theory, this allows us to describe in full the cohomology $R$-algebra $H^*(\text{Conf}(K_m,2);R)$ for any commutative unital ring $R$. As an application we prove that, although $\text{Conf}(K_m,2)$ is outside the "stable" regime, all its topological complexities are maximal possible when $m\geq4$.

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arXiv2022

Cohomology ring of tree braid groups and exterior face rings

For a tree $T$ and a positive integer $n$, let $B_nT$ denote the $n$-strand braid group on $T$. We use discrete Morse theory techniques to show that the cohomology ring $H^*(B_nT)$ is encoded by an explicit abstract simplicial complex $K_nT$ that measures $n$-local interactions among essential vertices of $T$. We show that, in many cases (for instance when $T$ is a binary tree), $H^*(B_nT)$ is the exterior face ring determined by $K_nT$.

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arXiv2021

Motion planning in polyhedral products of groups and a Fadell-Husseini approach to topological complexity

We compute the topological complexity of a polyhedral product $\mathcal{Z}$ defined in terms of an LS-logarithmic family of locally compact connected CW topological groups. The answer is given by a combinatorial formula that involves the LS category of the polyhedral-product factors. As a by-product, we show that the Iwase-Sakai conjecture holds true for $\mathcal{Z}$. The proof methodology uses a Fadell-Husseini viewpoint for the monoidal topological complexity (MTC) of a space, which, under mild conditions, recovers Iwase-Sakai's original definition. In the Fadell-Husseini context, the stasis condition -- MTC's raison d'être -- can be encoded at the covering level. Our Fadell-Husseini-inspired definition provides an alternative to the MTC variant given by Dranishnikov, as well as to the ones provided by García-Calcines, Carrasquel-Vera and Vandembroucq in terms of relative category.

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arXiv2021

On the homotopy type of complexes of graphs with bounded domination number

Let $D_{n,γ}$ be the complex of graphs on $n$ vertices and domination number at least $γ$. We prove that $D_{n,n-2}$ has the homotopy type of a finite wedge of 2-spheres. This is done by using discrete Morse theory techniques. Acyclicity of the needed matching is proved by introducing a relativized form of a well known method for constructing acyclic matchings on suitable chunks of simplices. Our approach allows us to extend our results to the realm of infinite graphs. In addition, we give evidence supporting the assertion that the homotopy equivalences $D_{n,n-1}\simeq\bigvee S^0$ and $D_{n,n-2}\simeq\bigvee S^2$ do not seem to generalize for $D_{n,γ}$ with $γ\leq n-3$.

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arXiv2021

Effectual Topological Complexity

We introduce the effectual topological complexity (ETC) of a $G$-space $X$. This is a $G$-equivariant homotopy invariant sitting in between the effective topological complexity of the pair $(X,G)$ and the (regular) topological complexity of the orbit space $X/G$. We study ETC for spheres and surfaces with antipodal involution, obtaining a full computation in the case of the torus. This allows us to prove the vanishing of twice the non-trivial obstruction responsible for the fact that the topological complexity of the Klein bottle is 4. In addition, this gives a counterexample to the possibility -- suggested in Pavešić's work on the topological complexity of a map -- that ETC of $(X,G)$ would agree with Farber's $TC(X)$ whenever the projection map $X\to X/G$ is finitely sheeted. We conjecture that ETC of spheres with antipodal action recasts the Hopf invariant one problem, and describe (conjecturally optimal) effectual motion planners.

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arXiv2021

Sectional category and The Fixed Point Property

For a Hausdorff space $X$, we exhibit an unexpected connection between the sectional number of the Fadell-Neuwirth fibration $π_{2,1}^X:F(X,2)\to X$, and the fixed point property (FPP) for self-maps on $X$. Explicitly, we demonstrate that a space $X$ has the FPP if and only if 2 is the minimal cardinality of open covers $\{U_i\}$ of $X$ such that each $U_i$ admits a continuous local section for $π_{2,1}^X$. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics.

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arXiv2020

On Lusternik-Schnirelmann category and topological complexity of no k-equal manifolds

We compute the Lusternik-Schnirelmann category and the topological complexity of no $k$-equal manifolds $M^{(k)}_d(n)$ for certain values of $d$, $k$ and $n$. This includes instances where $M^{(k)}_d(n)$ is known to be rationally non-formal. The key ingredient in our computations is the knowledge of the cohomology ring $H^*(M^{(k)}_d(n))$ as described by Dobrinskaya and Turchin. A fine tuning comes from the use of obstruction theory techniques.

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