Topological Complexity of wedges
We prove the formula \begin{equation*} TC(X\vee Y)=\max\{TC(X),TC(Y),cat(X\times Y)\} \end{equation*} for the topological complexity of the wedge $X\vee Y$.
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arXiv subjects
Cesar A. Ipanaque Zapata
Publications and source records attributed to Cesar A. Ipanaque Zapata.
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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Multitasking collision-free motion planning algorithms in Euclidean spaces
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.
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Sequential collision-free optimal motion planning algorithms in punctured Euclidean spaces
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.
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Collision-free spatial motion of rigid bodies via topological complexity
The Topological complexity a la Farber $\text{TC}(-)$ is a homotopy invariant which have interesting applications in Robotics, specifically, in the robot motion planning problem. In this work we calculate the topological complexity of the configuration space of $k$ distinct rigid bodies without collisions in $\mathbb{R}^d$, for $d=2,3$. Furthermore, we present optimal algorithms which can be used in designing practical systems controlling motion of many rigid bodies moving in space without collisions. The motion planning algorithms we present in this work are easily implementable in practice.
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Category and Topological Complexity of the configuration space $F(G\times \mathbb{R}^n,2)$
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 $G\times\mathbb{R}^n$ and apply the results to the planar and spatial motion of two rigid bodies in $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively.
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Lusternik-Schnirelmann category of the configuration space of complex projective space
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. Another similar invariant is the topological complexity $TC(X)$ (a la Farber) which has interesting applications in Robotics, specifically, in the robot motion planning problem. In this paper we calculate the Lusternik-Schnirelmann category and as a consequence we calculate the topological complexity of the two-point ordered configuration space of $\mathbb{CP}^n$ for every $n\geq 1$.
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Collision-free motion planning on manifolds with boundary
This paper concerns the study of the homotopy type of the ordered configuration space for manifolds with boundary and as an application we will study the collision free motion planning problem on manifolds with boundary.
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