Symmetric motion planning

Farber, Michael; Grant, Mark

doi:10.1090/conm/438/08447

Cited by 28 publications

(57 citation statements)

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“…Weights of cohomology classes with respect to a fibration. In this Section we recall the definition of weight of a cohomology class with respect to a fibration from [FG1]. We also give an alternative characterisation of weight in terms of fibred joins (Proposition 1), and show that classes with high category weight may lead to classes with high weight with respect to the path fibration π X (Theorem 3).…”

Section: Grantmentioning

confidence: 99%

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: Grantmentioning

confidence: 99%

Topological complexity of motion planning and Massey products

Grant

2009

Algebraic Topology - Old and New

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“…It is also Z=2-equivariant, in view of the last condition in (12). Therefore, it induces a corresponding (continuous) section x s i of the fibration ev 2 over the image of U i under the canonical (open) projection P r P r P r !…”

Section: Symmetric Axial Maps and Symmetric Tcmentioning

confidence: 98%

“…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 / !…”

Section: Definition 11mentioning

confidence: 99%

“….r / 3 5 5 9 9 9 9 17 17 19 23 23 This situation contrasts with the fact, proved in Section 6, that the first inequality in (9) becomes an equality for all complex projective spaces. Actually, in view of the main result in [20], there is even a (weird) possibility that the left hand side of (11) might turn out to be an unbounded function of r (compare to [12,Example 28]). Other situations with a behavior resembling that in (11) are given by spheres: according to [9; 12], TC S .S r / TC.S r / D 0 for n even, whereas TC S .S r / TC.S r / D 1 for n odd.…”

Section: Main Proofmentioning

confidence: 99%

“…S m satisfying the relations (17) z.!x; y/ D z.x; !y/ and z. x; y/ D z.x; y/ for x; y 2 S 2nC1 and ! 2 Z=2 e S 1 -these correspond to the first group of conditions in (12). Our first objective is to indicate how the slight variation in (18) below of the obvious symmetrization of these conditions describes the Euclidean embedding dimension for (arbitrary-torsion) lens spaces.…”

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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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Topological complexity of symplectic manifolds

Grant

Mescher

2019

Math. Z.

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We prove that the topological complexity of every symplectically atoroidal manifold is equal to twice its dimension. This is the analogue for topological complexity of a result of Rudyak and Oprea, who showed that the Lusternik-Schnirelmann category of a symplectically aspherical manifold equals its dimension. Symplectically hyperbolic manifolds are symplectically atoroidal, as are symplectically aspherical manifolds whose fundamental group does not contain free abelian subgroups of rank two. Thus we obtain many new calculations of topological complexity, including iterated surface bundles and symplectically aspherical manifolds with hyperbolic fundamental groups.

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Symmetric motion planning

Cited by 28 publications

References 18 publications

Topological complexity of motion planning and Massey products

Topological complexity of motion planning and Massey products

Symmetric topological complexity of projective and lens spaces

Topological complexity of symplectic manifolds

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