Robot motion planning, weights of cohomology classes, and cohomology operations

Farber, Michael; Grant, Mark

doi:10.1090/s0002-9939-08-09529-4

Cited by 35 publications

(61 citation statements)

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“…Fadell and Husseini achieved this in the case of category weight, using stable cohomology operations ( [FH] Theorem 3.12, see also Corollary 4.7 of [Rud]). An analogous result for TC was obtained by the authors in [FG2], where stable cohomology operations are used to find indecomposable zero-divisors z ∈ H * (X × X) with wgt π X (z) > 1, thus allowing the computation of TC of various lens spaces. Rudyak has shown ( [Rud], Corollary 4.6) that if u ∈ H * (X) is a Massey product then wgt(u) > 1 (the definition of Massey's triple product will be recalled in Section 3).…”

Section: Proof As Is Shown In Proposition 34 Ofsupporting

confidence: 53%

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Topological complexity of motion planning and Massey products

Grant

2009

Algebraic Topology - Old and New

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Abstract. We employ Massey products to find sharper lower bounds for the Schwarz genus of a fibration than those previously known. In particular we give examples of non-formal spaces X for which the topological complexity TC(X) (defined to be the genus of the free path fibration on X) is greater than the zero-divisors cup-length plus one.1. Introduction. Motion planning is a fundamental area of research in Robotics. A motion planning algorithm for a given mechanical system S is a function which assigns to each ordered pair (A, B) of physical states of S a continuous motion of S starting at A and ending at B. We may regard the admissible physical states of S as being parameterised by the points of a topological space X (the configuration space of the system) such that motions of the system correspond to continuous paths γ : [0, 1] = I → X. A motion planning algorithm for the system is then a section s : X × X → X I (not necessarily continuous) of the free path fibration

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Section: Proof As Is Shown In Proposition 34 Ofsupporting

confidence: 53%

“…In a recent paper of Farber and the author [FG2], stable cohomology operations are utilised to obtain sharper lower bounds for TC than the zero-divisors cup-length. In this article we investigate the effects of Massey products on topological complexity.…”

Section: Grantmentioning

confidence: 99%

Topological complexity of motion planning and Massey products

Grant

2009

Algebraic Topology - Old and New

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“…A number of properties of topological complexity and symmetric topological complexity can be found in [Fa03,Fa06,Fa08,FG07,FG08,FY04]. The papers [FTY03,GL09] identify these concepts in the case of real projective spaces as their immersion and embedding dimensions, respectively.…”

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confidence: 99%

Higher topological complexity and its symmetrization

Basabe

González

Rudyak

et al. 2014

Algebr. Geom. Topol.

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Abstract. We develop the properties of the n-th sequential topological complexity TC n , a homotopy invariant introduced by the third author as an extension of Farber's topological model for studying the complexity of motion planning algorithms in robotics. We exhibit close connections of TC n (X) to the Lusternik-Schnirelmann category of cartesian powers of X, to the cup-length of the diagonal embedding X → X n , and to the ratio between homotopy dimension and connectivity of X. We fully compute the numerical value of TC n for products of spheres, closed 1-connected symplectic manifolds, and quaternionic projective spaces. Our study includes two symmetrized versions of TC n (X). The first one, unlike Farber-Grant's symmetric topological complexity, turns out to be a homotopy invariant of X; the second one is closely tied to the homotopical properties of the configuration space of cardinality-n subsets of X. Special attention is given to the case of spheres.

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“…In fact, we extend Farber-Grant's result to the first case outside the high-torsion range by combining the techniques in [13] with the Z=m-biequivariant map characterization of TC.L 2nC1 .m// discussed at the beginning of Section 5.2. The result arose from an e-mail exchange, dating back to mid 2007, between the first author and Professor Farber concerning the results in [13]. Example 5.9 It is well known that the highest power of 2 dividing 2n n is˛.n/, the number of ones in the dyadic expansion of n. In particular, TC.L 2nC1 .2 e // D 4n C 2 for e >˛.n/.…”

Section: (Nonsymmetric) Tc Of High-torsion Lens Spacesmentioning

confidence: 80%

Symmetric topological complexity of projective and lens spaces

González

Landweber

2009

Algebr. Geom. Topol.

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For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even-torsion lens spaces and complex projective spaces are discussed. 55M30, 57R40Dedicated to the memory of Bob Stong Main resultThe Euclidean immersion and embedding questions for projective spaces were topics of intense research during the beginning of the second half of the last century. In the case of real projective spaces, the immersion problem has recently received a fresh push, partly in view of a surprising reformulation in terms of a basic concept arising in robotics, namely, the motion planning problem of mechanical systems. In more detail, Farber, Tabachnikov and Yuzvinsky [14] showed that for r ¤ 1; 3; 7, the immersion dimension of P r -the r -dimensional real projective space-agrees with TC.P r / 1, one unit less than the topological complexity of P r (see Definition 1.1 and Theorem 4.2 below). In this paper we accomplish a completely analogous goal by connecting the Euclidean embedding dimension of P r with Farber-Grant's notion of symmetric motion planning. Before stating our main results, we recall the relevant definitions. Definition 1.1 The topological complexity of a space X , TC.X /, is defined as the genus of the endpoints evaluation map evW P .X / ! X X , where P .X / is the free path space X OE0;1 with the compact-open topology.TC.X / is a homotopy invariant of X . Thinking of X as the space of configurations of a given mechanical system, TC.X / gives a measure of the topological instabilities in a motion planning algorithm for X -a perhaps discontinuous (but global) section of the map ev. We refer the reader to Farber [10] for a very useful survey of results in this area, and to the Farber's book [11] for a thorough introduction to the new mathematical discipline of topological robotics.We now come to the main definition (introduced and explored by Farber and Grant [12]). For a topological space X , let X be the diagonal in X X and ev 1 W P 1 .X / ! X X X be the restriction of the fibration ev in Definition 1.1. Thus P 1 .X / is the subspace of P .X / consisting of paths W OE0; 1 ! X with .0/ ¤ .1/. Note that ev 1 is a Z=2-equivariant map, where Z=2 acts freely on both P 1 .X / and X X X , by running a path backwards in the former, and by switching coordinates in the latter. Let P 2 .X / and B.X; 2/ denote the corresponding orbit spaces, and let ev 2 W P 2 .X / ! B.X; 2/ denote the resulting fibration.Definition 1.2 With the above conditions, the symmetric topological complexity of X , TC S .X /, is defined by TC S .X / D genus.ev 2...

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Robot motion planning, weights of cohomology classes, and cohomology operations

Cited by 35 publications

References 10 publications

Topological complexity of motion planning and Massey products

Topological complexity of motion planning and Massey products

Higher topological complexity and its symmetrization

Symmetric topological complexity of projective and lens spaces

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