scholarly journals Hopf invariants for sectional category with applications to topological robotics

2019 ◽  
Author(s):  
Jesús González ◽  
Mark Grant ◽  
Lucile Vandembroucq
Keyword(s):  
Topological Complexity ◽  
Cell Complexes ◽  
Lusternik Schnirelmann ◽  
Hopf Invariants

Abstract We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular, we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied to the study of Farber’s topological complexity for two-cell complexes, as well as to the construction of a counterexample to the analogue for topological complexity of Ganea’s conjecture on Lusternik–Schnirelmann category.

scholarly journals Topological complexity of collision-free motion planning on surfaces

2010 ◽  
Vol 147 (2) ◽  
pp. 649-660 ◽  
Author(s):  
Daniel C. Cohen ◽  
Michael Farber
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Main Tool ◽  
Orientable Surface ◽  
Configuration Spaces ◽  
Topological Complexity ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Lusternik Schnirelmann ◽  
Mixed Hodge Structures

AbstractThe topological complexity$\mathsf {TC}(X)$is a numerical homotopy invariant of a topological spaceXwhich is motivated by robotics and is similar in spirit to the classical Lusternik–Schnirelmann category ofX. Given a mechanical system with configuration spaceX, the invariant$\mathsf {TC}(X)$measures the complexity of motion planning algorithms which can be designed for the system. In this paper, we compute the topological complexity of the configuration space ofndistinct ordered points on an orientable surface, for both closed and punctured surfaces. Our main tool is a theorem of B. Totaro describing the cohomology of configuration spaces of algebraic varieties. For configuration spaces of punctured surfaces, this is used in conjunction with techniques from the theory of mixed Hodge structures.

scholarly journals On the topological complexity of aspherical spaces

2018 ◽  
Vol 12 (02) ◽  
pp. 293-319 ◽  
Author(s):  
Michael Farber ◽  
Stephan Mescher
Keyword(s):  
Spectral Sequence ◽  
Cohomology Class ◽  
Hyperbolic Group ◽  
Cohomological Dimension ◽  
Topological Complexity ◽  
Canonical Class ◽  
Lusternik Schnirelmann

The well-known theorem of Eilenberg and Ganea [Ann. Math. 65 (1957) 517–518] expresses the Lusternik–Schnirelmann category of an Eilenberg–MacLane space [Formula: see text] as the cohomological dimension of the group [Formula: see text]. In this paper, we study a similar problem of determining algebraically the topological complexity of the Eilenberg–MacLane spaces [Formula: see text]. One of our main results states that in the case when the group [Formula: see text] is hyperbolic in the sense of Gromov, the topological complexity [Formula: see text] either equals or is by one larger than the cohomological dimension of [Formula: see text]. We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class (as defined by Costa and Farber) via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group, we establish a vanishing property of this spectral sequence which leads to the main result.

scholarly journals Homotopic distance between maps

2021 ◽  
pp. 1-21
Author(s):  
E. MACÍAS–VIRGÓS ◽  
D. MOSQUERA–LOIS
Keyword(s):  
General Notion ◽  
Topological Complexity ◽  
Lusternik Schnirelmann

Abstract We show that well-known invariants like Lusternik–Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.

scholarly journals Min-Max Theory for Cell Complexes

2020 ◽  
Vol 27 (03) ◽  
pp. 447-454
Author(s):  
Lacey Johnson ◽  
Kevin Knudson
Keyword(s):  
Morse Function ◽  
Critical Values ◽  
Discrete Version ◽  
Mountain Pass Lemma ◽  
Smooth Functions ◽  
Alternate Proof ◽  
Cell Complexes ◽  
Lusternik Schnirelmann ◽  
Regular Cell Complex ◽  
Discrete Morse Function

In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. We introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik–Schnirelmann theorem.

scholarly journals A Randomized Greedy Algorithm for Piecewise Linear Motion Planning

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2358
Author(s):  
Carlos Ortiz ◽  
Adriana Lara ◽  
Jesús González ◽  
Ayse Borat
Keyword(s):  
Motion Planning ◽  
Piecewise Linear ◽  
Randomized Algorithm ◽  
Robot Motion ◽  
Topological Complexity ◽  
Robot Motion Planning ◽  
Automated Guided Vehicle ◽  
Linear Motion ◽  
Lusternik Schnirelmann ◽  
Robust To Noise

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R, and outputs an explicit system of piecewise linear motion planners for R. The algorithm is designed in such a way that the cardinality of the output is probabilistically close (with parameters chosen by the user) to the minimal possible cardinality.This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber’s topological complexity TC. Besides its relevance toward technological applications, our work reveals that, unlike other discrete approaches to TC, the SC model can recast Farber’s invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Macías-Virgós and Mosquera-Lois’ notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra.

scholarly journals CATEGORY AND TOPOLOGICAL COMPLEXITY OF THE CONFIGURATION SPACE

2019 ◽  
Vol 100 (3) ◽  
pp. 507-517
Author(s):  
CESAR A. IPANAQUE ZAPATA
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Rigid Bodies ◽  
Robot Motion ◽  
Planning Problem ◽  
Topological Complexity ◽  
Robot Motion Planning ◽  
Spatial Motion ◽  
Lusternik Schnirelmann ◽  
Homotopy Invariants

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.

scholarly journals Topological complexity of real Grassmannians

2021 ◽  
pp. 1-17
Author(s):  
Petar Pavešić
Keyword(s):  
Motion Planning ◽  
Zero Divisor ◽  
Cohomology Ring ◽  
Detailed Knowledge ◽  
Topological Complexity ◽  
Grassmann Manifolds ◽  
Linear Subspaces ◽  
Lusternik Schnirelmann

We use some detailed knowledge of the cohomology ring of real Grassmann manifolds G k (ℝ n ) to compute zero-divisor cup-length and estimate topological complexity of motion planning for k-linear subspaces in ℝ n . In addition, we obtain results about monotonicity of Lusternik–Schnirelmann category and topological complexity of G k (ℝ n ) as a function of n.

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