arXiv subjects
Publications and source records attributed to Cesar A. Ipanaque Zapata.
We develop practical tools for analyzing the configuration set $F(G,k)$ of $k\geq 2$ distinct elements in a group $G$. We apply our results to design homogeneous linear systems in $\mathbb{F}_q$ that admit nontrivial solutions. Furthermore, we study the connectivity of Cayley graphs of the form $\mathrm{Cay}(G^k,F(G,k))$. In addition, we consider the configuration set of $k\geq 2$ distinct non-identity elements in $G$.
We introduce the concept of injective category number $\text{IC}(f)$ for a continuous map $f\colon X\to~Y$, and present fundamental results concerning this numerical invariant. The value $\text{IC}(f)$ quantifies the \aspas{complexity} or \aspas{categorical structure} underlying the question: under what conditions is $f$ injective? More precisely, $\text{IC}(f)$ is the smallest positive integer $\ell$ such that $X$ can be covered by $\ell$ open subsets $U_1,\ldots,U_\ell$, with each restriction map $f_{\mid U}:U\to Y$ being injective. For instance, we examine the behaviour of $\text{IC}(f)$ under pullbacks and compositions of maps. In addition, we provide a cohomological lower bound for $\text{IC}(f)$. When $f$ has a finite number of multiple points, we express $\text{IC}(f)$ in terms of these points of non-injectivity. In the case that $f$ is the quotient map $\mathfrak{q}^X:X\to X/G$, where $X$ is a metric free $G$-space, we provide a lower bound for the injective category of $\mathfrak{q}^X$ in terms of the $2$-th index, $\text{ind}_2(X,G)$. When $G=\mathbb{Z}_2$, this lower bound is shown to be sharp. These results link a classical problem in Borsuk-Ulam theory to contemporary research developments in the study of injective category numbers.
For a Hausdorff space $Y$, a topological space $X$ and a map $g:X\to Y$, we present a connection between the relative sectional number of the first coordinate projection $π_{2,1}^Y:F(Y,2)\to Y$ with respect to $g$, and the coincidence property (CP) for $(X,Y;g)$, where $F(Y,2)$ stands for the ordered configuration space of $2$ distinct points on $Y$, and $(X,Y;g)$ has the coincidence property (CP) if, for every map $f:X\to Y$, there is a point $x$ of $X$ such that $f(x)=g(x)$. Explicitly, we demonstrate that $(X,Y;g)$ has the CP if and only if 2 is the minimal cardinality of open covers $\{U_i\}_{1\leq i\leq n}$ of $X$ such that each $U_i$ admits a local lifting for $g$ with respect to $π_{2,1}^Y$. This characterization connects a standard problem in coincidence theory to current research trends in sectional category and topological robotics. Motivated by this connection, we introduce the notion of relative topological complexity of a map.
We present the notion of injective hom-complexity, leading to a connection between the covering number of a group and the sectional number of a group homomorphism, and provide estimates for computing this invariant.
We present the notion of facet-complexity, $\text{C}(\mathsf{L};\mathsf{K})$, for two simplicial complexes $\mathsf{L}$ and $\mathsf{K}$, along with basic results for this numerical invariant. This invariant $\text{C}(\mathsf{L};\mathsf{K})$ quantifies the \aspas{complexity} of the following question: When does there exist a facet simplicial map $\mathsf{L}\to \mathsf{K}$? A facet simplicial map is a simplicial map that preserves non-unitary facets. Likewise, we introduce the notion of injective facet-complexity, $\text{IC}(\mathsf{L};\mathsf{K})$. These invariants generalize the notion of (injective) hom-complexity between graphs, recently introduced by Zapata et al. We demonstrate a triangular inequality for (injective) facet-complexity and show that it is a simplicial complex invariant. Additionally, these invariants provide an obstruction to the existence of facet simplicial maps. We explore the sub-additivity of (injective) facet-complexity and we present a lower bound in terms of the chromatic number. Moreover, we provide an upper bound for $\mathrm{C}(\mathsf{L};\mathsf{H})$ in terms of the number of facets of $L$. Finally, we establish a formula for $\mathrm{IC}(\mathsf{L};\mathsf{K})$ when $\mathsf{L}$ is a pure simplicial complex and $K$ is a complete simplicial complex.
We explore the notion of sectional number of a group homomorphism, leading to a generalization of the covering number of a group, and present several characterizations when the sectional number is finite, providing estimates for computing this invariant.
We present the notion of hom-complexity, $\text{C}(G;H)$, for two graphs $G$ and $H$, along with basic results for this numerical invariant. This invariant $\text{C}(G;H)$ is a number that measures the \aspas{complexity} of the question: when is there a homomorphism $G\to H$? More precisely, $\text{C}(G;H)$ is the least positive integer $k$ such that there are $k$ different subgraphs $G_j$ of $G$ such that $G=G_1\cup\cdots\cup G_k$, and for each $G_j$, there is a homomorphism $G_j\to H$. Likewise, we introduce the notion of injective hom-complexity, $\text{IC}(G;H)$. The (injective) hom-complexity is a graph invariant. Additionally, these invariants can be used to show the nonexistence of homomorphisms. We explore the sub-additivity of (injective) hom-complexity and study products. We describe bounds for the hom-complexity in terms of chromatic number $χ$ and clique number $ω$. We provide the formula \[\text{C}(G;H)=\lceil\log_{χ(H)}χ(G)\rceil\] whenever $ω(H)=χ(H)$. For example, we obtain $\text{C}(G;K_\ell)=\lceil\log_{\ell}χ(G)\rceil$. Moreover, we discuss a connection between the (injective) hom-complexity and several well-known covering numbers. For instance, we provide a lower bound for the clique covering number in terms of the injective hom-complexity. Additionally, we show that the hom-complexity $\mathrm{C}(G;K_{\ell})$ coincides with the $\ell$-particity $β_\ell(G)$ of $G$, and the hom-complexity $\mathrm{C}(K_n;K_{2})$ coincides with the bipartite dimension $\mathrm{d}(K_n)$ of $K_n$. As a consequence, we recover the well-known formulas $β_\ell(G)=\lceil\log_{\ell}χ(G)\rceil$ and $\mathrm{d}(K_n)=\lceil\log_{2}n\rceil$.
In this work, we show that the étale sectional number (\text{Étale-sec$(-)$}), i.e., the sectional number in the category of topological spaces with the étale quasi Grothendieck topology (as defined in arXiv:2410.22515), is either 1 or infinity. Specifically, given a continuous map $f:X\to Y$, we demonstrate that \[\text{Étale-sec$(f)$}=\begin{cases} 1,&\hbox{ if $f$ is locally sectionable,} \infty,&\hbox{ if $f$ is not locally sectionable.} \end{cases} \] Additionally, for a path-connected space $X$, the étale topological complexity satisfies \[\text{TC}_{\text{étale}}(X)=\begin{cases} 1,&\hbox{ if $X$ is locally contractible,} \infty,&\hbox{ if $X$ is not locally contractible.} \end{cases} \] These results provide a way to understand the \aspas{complexity} of maps and spaces within the context of the étale quasi Grothendieck topology, a structure that considers local behavior of maps and spaces. The classification into values of 1 or infinity reflects a dichotomy in the local geometric structure of the map or space, with the presence or absence of local sections or contractibility significantly influencing the outcome.
The genus of a fibration was introduced by Schwarz in 1962. Given a continuous map $g:A\to B$, the usual sectional number $\text{sec}_u(g)$ is the least integer~$m$ such that $B$ can be covered by $m$ open subsets, each of which admits a local section of~$g$. Likewise, the sectional category $\text{secat}(g)$ is the least integer~$m$ such that $B$ can be covered by $m$ open subsets, each of which admits a local homotopy section of~$g$. In the case that $g$ is a fibration, the usual sectional number and the sectional category of $g$ coincide with the Schwarz's genus of $g$. In this paper, we introduce a notion of sectional number for a morphism in a category with covers, which extends the usual sectional number and sectional category (and, of course, Schwarz's notion). We study the invariance property, the behaviour under weak pullbacks and continuous functors, and present upper bounds in terms of a notion of LS category and projective covering number for an object. In addition, we introduce a notion of multiplicative cohomology theory on a category with covers, and we use it to present a cohomological lower bound for the sectional number. Several examples of this invariant are presented to support this theory. For instance, our sectional number yields a family of new numerical invariants in Top; we also express the twin prime conjecture and Goldbach's conjecture in terms of sectional numbers in the category of sets. Our unified concept is relevant in this new setting because it provides a foundation for sectional theory in category theory and builds bridges between new areas in mathematics.
In this paper we discover a connection between the Milnor fibration theory and current research trends in topological robotics. The configuration space and workspace are often described as subspaces of some Euclidean spaces. The work map is a continuous map which assigns to each state of the configuration space the position of the end-effector at that state. The tasking planning problem consists in study the complexity and design of algorithms controlling the task performed by the robot manipulator. We study the tasking planning problem for the Milnor fibration of analytic map germs. We see that the tasking planning algorithms strongly depend of the Milnor fibration theorem. We conjecture that the topological complexity of Milnor fibration coincides with the topological complexity of its base.
In this paper, for fibrations $p:E\to B$, $p':E'\to B$ over $B$ and a free involution $τ:E\to E$ which satisfy the equality $p\circτ=p$, we discover a connection between the sectional category of the double covers $q:E\to E/τ$ and $q^Y:F_{p'}(E',2)\to F_{p'}(E',2)/\mathbb{Z}_2$ from the $2$-ordered fibre-wise configuration space $F_{p'}(E',2)$ to its unordered quotient $ F_{p'}(E',2)/\mathbb{Z}_2$, and the parametrized Borsuk-Ulam property (PBUP) for the triple $\left((p,τ);p'\right)$. Explicitly, we demonstrate that the triple $\left((p,τ);p'\right)$ satisfies the PBUP if the sectional category of $q$ is bigger than the sectional category of $q^{p'}$. This property connects a standard problem in parametrized Borsuk-Ulam theory to current research trends in sectional category. As applictions of our results, we explore the PBUP for $E, E'$ one of the following fibrations: trivial fibration, Hopf fibration and the Fadell-Neuwirth fibration.
A classic problem in analysis is to solve nonlinear equations of the form \begin{equation*} F(x)=0, \end{equation*} where $F:D^n\to \mathbb{R}^m$ is a continuous map of the closed unit disk $D^n\subset\mathbb{R}^n$ in $\mathbb{R}^m$. A topological technique, which exists in the literature, for the existence of solutions of nonlinear equations is the topological degree theory. In this work, we will use the category of a map theory to solve the problem of existence of solutions of nonlinear equations. This theory, as we will show in this work, provides an alternative topological technique to study nonlinear equations.
For a Hausdorff space $X$, a free involution $τ:X\to X$ and a Hausdorff space $Y$, we discover a connection between the sectional category of the double covers $q:X\to X/τ$ and $q^Y:F(Y,2)\to D(Y,2)$ from the ordered configuration space $F(Y,2)$ to its unordered quotient $D(Y,2)=F(Y,2)/Σ_2$, and the Borsuk-Ulam property (BUP) for the triple $\left((X,τ);Y\right)$. Explicitly, we demonstrate that the triple $\left((X,τ);Y\right)$ satisfies the BUP if the sectional category of $q$ is bigger than the sectional category of $q^Y$. This property connects a standard problem in Borsuk-Ulam theory to current research trends in sectional category. As an application of our results, we show that the index of $(X,τ)$ coincides with the sectional category of the quotient map $q:X\to X/τ$ minus 1 for any paracompact space $X$. In addition, we present some new results relating Borsuk-Ulam theory and sectional category.
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.
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.
In this paper, we introduce the notion of transversal topological complexity (TTC) for a smooth manifold $X$ with respect to a submanifold of codimension 1 together with basic results about this numerical invariant. In addition, we present several examples of explicit transversal algorithms.
In this paper we present new results about the topology of the Milnor fibrations of analytic function-germs with a special attention to the topology of the fibers. In particular, we provide a short review on the existence of the Milnor fibrations in the real and complex cases. This allows us to compare our results with the previous ones.
In this work we will review the notion of topological complexity, introduced by Michael Farber in 2003. We will use this theory of topological complexity to solve the motion planning problem of a mobile robot that navigates in the Euclidean plane avoiding colliding with an obstacle. Specifically, we calculate topological complexity and design explicit algorithms.