Topological complexity of motion planning in projective product spaces

González, Jesús; Grant, Mark; Torres-Giese, Enrique; Xicoténcatl, Miguel A.

doi:10.2140/agt.2013.13.1027

Cited by 9 publications

(6 citation statements)

References 24 publications

(49 reference statements)

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“…Also by analogy with the non-equivariant case, we have the following product formula, which was first stated as [28,Theorem 4.2] under the assumption that X and Y are smooth G-manifolds. Theorem 3.5.…”

Section: Equivariant Topological Complexitymentioning

confidence: 99%

Equivariant topological complexities

Ángel¹,

Colman²

2018

Topological Complexity and Related Topics

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Many mechanical systems have configuration spaces that admit symmetries. Mathematically, such symmetries are modelled by the action of a group on a topological space. Several variations of topological complexity have emerged that take symmetry into account in various ways, either by asking that the motion planners themselves admit compatible symmetries, or by exploiting the symmetry to motion plan between functionally equivalent configurations. We will survey the main definitions due to Colman-Grant, Lubawski-Marzantowicz, B laszczyk-Kaluba and Dranishnikov, and some related notions. We conclude with a short list of open problems.Example 1.1. Consider a planar mechanism (such as a robot arm), one component of which is anchored to a point in the plane. Denote its configuration space by Y . Now add an extra revolute joint, giving one more degree of freedom. We imagine creating a spatial mechanism by basing the anchor of the arm to a rotating platform or circular track in 3-space. The configuration space of this new mechanism is the topological product X := S 1 × Y , where S 1 is the unit circle.

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Section: Equivariant Topological Complexitymentioning

confidence: 99%

Equivariant topological complexities

Ángel¹,

Colman²

2018

Topological Complexity and Related Topics

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“…Such a guess is formalized in [9, Theorem 3.8]-even if no sphere splits off P (n 1 ,n 2 ,...,nr) as a cartesian factor. In this section we adapt the viewpoint in [9] in order to obtain similar estimates for the category and the topological complexity of complex-projective and lens product spaces.…”

Section: Category and Topological Complexitymentioning

confidence: 99%

Complex-projective and lens product spaces

González

Velasco

2014

Bol. Soc. Mat. Mex.

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Abstract. Let t be a positive integer. Following work of D. M. Davis, we study the topology of complex-projective product spaces, i.e. quotients of cartesian products of odd dimensional spheres by the diagonal S 1 -action, and of the t-torsion lens product spaces, i.e. the corresponding quotients when the action is restricted to the t th roots of unity. For a commutative complex-oriented cohomology theory h * , we determine the h * -cohomology ring of these spaces (in terms of the t-series for h * , in the case of t-torsion lens product spaces). When h * is singular cohomology with mod 2 coefficients, we also determine the action of the Steenrod algebra. We show that these spaces break apart after a suspension as a wedge of desuspensions of usual stunted complex projective (t-torsion lens) spaces. We estimate the category and topological complexity of complex-projective and lens product spaces, showing in particular that these invariants are usually much lower than predicted by the usual dimensional bounds. We extend Davis' analysis of manifold properties such as immersion dimension, (stable-)span, and (stable-)parallelizability of real projective product spaces to the complex-projective and lens product cases.

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“…This space can be considered as a generalization of real projective space but is not in general product of projective spaces. The topological complexity and some bounds of these spaces have been initiated in [17]. The improvement of this study to finalize the estimating problem about the topological complexity and the Lusternik-Schnirelmann category of projective product spaces has been included in [16].…”

Section: Introductionmentioning

confidence: 99%

Explicit motion planning in digital projective product spaces

Fişekci¹,

Karaca²

2021

Preprint

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We introduce the digital projective product spaces based on Davis' projective product spaces. We determine an upper bound for the digital LS-category of the digital projective product spaces. In addition, we obtain an upper bound for the digital topological complexity of these spaces. We prove the relation between the digital topological complexity of the digital projective product spaces and sum of the digital topological complexity of the digital projective space by associating with the first digital sphere and the digital topological complexity of the remaining digital spheres through an explicit motion planning construction, which shows digital perspective validity of the results given by S. Fişekci and L. Vandembroucq.We apply our outcomes on specific spaces in order to be more clear.

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Topological complexity of motion planning in projective product spaces

Cited by 9 publications

References 24 publications

Equivariant topological complexities

Equivariant topological complexities

Complex-projective and lens product spaces

Explicit motion planning in digital projective product spaces

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