Papers by Cesar A. Ipanaque Zapata
arXiv (Cornell University), Aug 22, 2022
In this paper we present new results about the topology of the Milnor fibrations of analytic func... more 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.
Bulletin of the Iranian Mathematical Society
For a Hausdorff space X, a free involution τ : X → X and a Hausdorff space Y , we discover a conn... more For a Hausdorff space X, a free involution τ : X → X and a Hausdorff space Y , we discover a connection between the sectional category of the double covers q : X → X/τ and q Y : F (Y, 2) → 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 ((X, τ); Y). Explicitly, we demonstrate that the triple ((X, τ); Y) 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 → X/τ minus 1 for any paracompact space X. In addition, we present some new results relating Borsuk-Ulam theory and sectional category.
arXiv (Cornell University), Mar 12, 2023
In this paper, we introduce the notion of transversal topological complexity (TTC) for a smooth m... more 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.
arXiv (Cornell University), Dec 6, 2022
The higher topological complexity of a space X, TC r (X), r = 2, 3,. . ., and the topological com... more The higher topological complexity of a space X, TC r (X), r = 2, 3,. . ., and the topological complexity of a map f , 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 , TC r,s (f), for 1 ≤ s ≤ r ≥ 2, 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 TC r,s under products and compositions of maps, as well as the dependence of TC r,s on r and s. We draw general estimates for TC r,s (f : X → Y) in terms of categorical invariants associated to X, Y and f. In particular, we describe within one the value of TC r,s in the case of the non-trivial double covering over real projective spaces, as well as for their complex counterparts.
Bulletin of the Brazilian Mathematical Society, New Series
In this paper we present new results about the topology of the Milnor fibrations of analytic func... more 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.
arXiv (Cornell University), Oct 1, 2022
For a Hausdorff space X, a free involution τ : X → X and a Hausdorff space Y , we discover a conn... more For a Hausdorff space X, a free involution τ : X → X and a Hausdorff space Y , we discover a connection between the sectional category of the double covers q : X → X/τ and q Y : F (Y, 2) → 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 ((X, τ); Y). Explicitly, we demonstrate that the triple ((X, τ); Y) 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 → X/τ minus 1 for any paracompact space X. In addition, we present some new results relating Borsuk-Ulam theory and sectional category.
O objetivo principal deste trabalho será apresentar um estudo topológico do Problema de Planejame... more O objetivo principal deste trabalho será apresentar um estudo topológico do Problema de Planejamento de Movimento de Robôs sem colisões. Especificamente, este trabalho trata da abordagem dos seguintes importantes problemas na área de Geometria e Topologia: (1) estudo do problema de planificação de movimento livre de colisões, para um número de espaços mecânicos importantes que aparecem na robótica. (2) estudo da complexidade topológica de (alguns) espaços mecânicos. (3) estudo dos variantes da complexidade topológica e seus problemas relacionados. (4) estudo da complexidade topológica para espaços asféricos. (5) entender a estrutura do anel de cohomologia de (alguns) espaços de configurações.
We prove the formula cat_G(X∨ Y) = max{cat_G(X), cat_G(Y)} for the equivariant category of the we... more We prove the formula cat_G(X∨ Y) = max{cat_G(X), cat_G(Y)} for the equivariant category of the wedge X∨ Y.
O objetivo principal deste trabalho será apresentar um estudo detalhado dos espaços de configuraç... more O objetivo principal deste trabalho será apresentar um estudo detalhado dos espaços de configurações. Dissertaremos sobre: espaços de configurações clássicos, invariância do bordo, espaço de configurações para superfícies, fibração de Fadell e Neuwirth e espaços de configurações do espaço Euclideano, da esfera e do espaço projetivo complexo.
We present optimal motion planning algorithms which can be used in designing practical systems co... more We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions. Our algorithms are optimal in a very concrete sense, namely, they have the minimal possible number of local planners. Our algorithms are motivated by those presented by Mas-Ku and Torres-Giese (as streamlined by Farber), and are developed within the more general context of the multitasking (a.k.a.~higher) motion planning problem. In addition, an eventual implementation of our algorithms is expected to work more efficiently than previous ones when applied to systems with a large number of moving objects.
For a Hausdorff space X, we exhibit an unexpected connection between the sectional number of the ... more 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)→ 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.
The Lusternik-Schnirelmann category cat and topological complexity TC are related homotopy invari... more The Lusternik-Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik-Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product G×R^n and apply the results to the planar and spatial motion of two rigid bodies in R^2 and R^3 respectively.
In robotics, a topological theory of motion planning was initiated by M. Farber. The multitasking... more In robotics, a topological theory of motion planning was initiated by M. Farber. The multitasking motion planning problem is new and its theoretical part via topological complexity has hardly been developed, but the concrete implementations are still non-existent, and in fact this work takes the first step in this last direction (producing explicit algorithms.) We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions between them and avoiding obstacles. Furthermore, we present the multitasking version of the algorithms.
Pesquimat, 2021
En este trabajo revisaremos la noción de complejidad topológica, introducida por Michael Farber e... more En este trabajo revisaremos la noción de complejidad topológica, introducida por Michael Farber en el 2003. Usaremos esta teoría de complejidad topológica para resolver el problema de planificación de movimiento de un robot móvil que navega en el plano euclidiano evitando colisionar con un obstáculo. Específicamente, calculamos la complejidad topológica y diseñamos algoritmos óptimos.
arXiv: Algebraic Topology, 2019
Consider a fibration \[p:E\to S^{p-1}\] with fiber $F$. We have the following natural question: U... more Consider a fibration \[p:E\to S^{p-1}\] with fiber $F$. We have the following natural question: Under what conditions does this fibration admit a cross-section? Our purpose is to discuss this problem when the fibration $p$ is the Milnor fibration $f_{\mid}:f^{-1}(S^{p-1}_{\delta})\cap D^{n}_{\epsilon}\to S^{p-1}_{\delta}$ with Milnor fiber $F_f$ and the Milnor fibration of arrangements $Q:\mathcal{M}(\mathcal{A})\to \mathbb{C}^\ast$ with fiber $F=F(\mathcal{A})$. Furthermore, we use our results to study the tasking planning problem for the Milnor fibration as a work map and we give the tasking algorithms.
ArXiv, 2021
We present optimal parametrised motion planning algorithms which can be used in designing practic... more We present optimal parametrised motion planning algorithms which can be used in designing practical systems controlling objects that move in Euclidean d-space, with d ≥ 2 even, without collisions and in the presence of two obstacles with unknown a priori positions. Our algorithms are optimal in a very concrete sense, namely, they have the minimal possible number of local planners.
arXiv: Algebraic Topology, 2017
Let $F(X,k)$ be the configuration spaces of ordered $k-$tuples of distinct points in the space $X... more Let $F(X,k)$ be the configuration spaces of ordered $k-$tuples of distinct points in the space $X$. Using the Fadell and Neuwirth's fibration, we prove that the configuration space $F(M,k)$ of certain topological manifolds $M$, is not contractible.
arXiv: Algebraic Topology, 2017
The Lusternik-Schnirelmann category $cat(X)$ is a homotopy invariant which is a numerical bound o... more The Lusternik-Schnirelmann category $cat(X)$ is a homotopy invariant which is a numerical bound on the number of critical points of a smooth function on a manifold. In this paper we calculate the Lusternik-Schnirelmann category of the configuration space of $2$ distinct points in Complex Projective $n-$space for all $n\geq 1$.
In this work we will review the notion of topological complexity, introduced by Michael Farber in... more 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.
We prove the formula \begin{equation*}cat_G(X\vee Y) = \max \{cat_G(X), cat_G(Y)\} \end{equation*... more We prove the formula \begin{equation*}cat_G(X\vee Y) = \max \{cat_G(X), cat_G(Y)\} \end{equation*} for the equivariant category of the wedge $X\vee Y$.
Uploads
Papers by Cesar A. Ipanaque Zapata