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.

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 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 On the topological complexity of Grassmann manifolds

2020 ◽  
Vol 70 (5) ◽  
pp. 1197-1210
Author(s):  
Vimala Ramani
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Zero Divisor ◽  
Flag Manifold ◽  
Topological Complexity ◽  
Grassmann Manifolds ◽  
Real Dimension ◽  
The Real ◽  
Rational Cohomology

AbstractWe prove that the topological complexity of a quaternionic flag manifold is half of its real dimension. For the real oriented Grassmann manifolds G͠n,k, 3 ≤ k ≤ [n/2], the zero-divisor cup-length of the rational cohomology of G͠n,k is computed in terms of n and k which gives a lower bound for the topological complexity of G͠n,k, TC(G͠n,k). When k = 3, it is observed in certain cases that better lower bounds for TC(G͠n,3) are obtained using ℤ2-cohomology.

scholarly journals Symmetric Bi-Skew Maps and Symmetrized Motion Planning in Projective Spaces

2018 ◽  
Vol 61 (4) ◽  
pp. 1087-1100
Author(s):  
Jesús González
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Motion Planning ◽  
Cohomology Ring ◽  
Projective Spaces ◽  
Closed Surface ◽  
Topological Complexity ◽  
Embedding Dimension ◽  
The Core

AbstractThis work is motivated by the question of whether there are spacesXfor which the Farber–Grant symmetric topological complexity TCS(X) differs from the Basabe–González–Rudyak–Tamaki symmetric topological complexity TCΣ(X). For a projective space${\open R}\hbox{P}^m$, it is known that$\hbox{TC}^S ({\open R}\hbox{P}^{m})$captures, with a few potential exceptional cases, the Euclidean embedding dimension of${\open R}\hbox{P}^{m}$. We now show that, for allm≥1,$\hbox{TC}^{\Sigma}({\open R}\hbox{P}^{m})$is characterized as the smallest positive integernfor which there is a symmetric${\open Z}_{2}$-biequivariant mapSm×Sm→Snwith a ‘monoidal’ behaviour on the diagonal. This result thus lies at the core of the efforts in the 1970s to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both TC numbers in the case of${\open R}\hbox{P}^{2^{e}}$fore≥1. In particular, this leaves the torusS1×S1as the only closed surface whose symmetric (symmetrized) TCS(TCΣ) invariant is currently unknown.

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 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 Topological complexity of motion planning in projective product spaces

2013 ◽  
Vol 13 (2) ◽  
pp. 1027-1047 ◽  
Author(s):  
Jesús González ◽  
Mark Grant ◽  
Enrique Torres-Giese ◽  
Miguel Xicoténcatl
Keyword(s):  
Motion Planning ◽  
Topological Complexity ◽  
Product Spaces
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