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31
Topological robotics: motion planning in projective spaces
 International Mathematical Research Notices
, 2003
"... In this paper, we study one of the most elementary problems of the topological robotics: rotation of a line, which is fixed by a revolving joint at a base point. One wants to ..."
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Cited by 20 (6 self)
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In this paper, we study one of the most elementary problems of the topological robotics: rotation of a line, which is fixed by a revolving joint at a base point. One wants to
Embedding rightangled Artin groups into graph braid groups
 Department of Mathematics, Miami University
"... Abstract. We construct an embedding of any rightangled Artin group G(∆) defined by a graph ∆ into a graph braid group. The number of strands required for the braid gropu is equal to the chromatic number of ∆. This construction yields an example of a hyperbolic surface subgroup embedded in a two str ..."
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Cited by 13 (4 self)
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Abstract. We construct an embedding of any rightangled Artin group G(∆) defined by a graph ∆ into a graph braid group. The number of strands required for the braid gropu is equal to the chromatic number of ∆. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group. Mathematics Subject Classification (2000): 20F36, 20F65, 05C25 1.
Higher topological complexity and homotopy dimension . . .
, 2012
"... Yu. Rudyak has recently extended Farber’s notion of topological complexity by defining, for n ≥ 2, the nth topological complexity TCn(X) of a pathconnected space X—Farber’s original notion is recovered for n = 2. In this paper we develop further the properties of this extended concept, relating i ..."
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Cited by 12 (3 self)
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Yu. Rudyak has recently extended Farber’s notion of topological complexity by defining, for n ≥ 2, the nth topological complexity TCn(X) of a pathconnected space X—Farber’s original notion is recovered for n = 2. In this paper we develop further the properties of this extended concept, relating it to the LusternikSchnirelmann category of cartesian powers of X, as well as to the cuplength of the diagonal embedding X ↪ → Xn. We compute the numerical values of TCn for products of spheres, closed 1connected symplectic manifolds (e.g. complex projective spaces), and quaternionic projective spaces. We explore the symmetrized version of the concept (TCSn(X)) and introduce a new symmetrization (TCΣn (X)) which is a homotopy invariant of X. We obtain a (conjecturally sharp) upper bound for TCSn(X) when X is a sphere. This is attained by introducing and studying the idea of cellular stratified spaces, a new concept that allows us to import techniques from the theory of hyperplane arrangements in order to construct finite CW complexes of the lowest possible dimension modelling, up to equivariant homotopy, configuration spaces of ordered distinct points on spheres— our models are in fact simplicial complexes. In particular, we show that the configuration
Robot motion planning, weights of cohomology classes, and cohomology operations
"... Abstract. The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC(X) of the configuration space X of the system. Previously known lower bounds for TC(X) use the structure of the cohomology algebra of X. In this paper we show how cohomology operations c ..."
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Abstract. The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC(X) of the configuration space X of the system. Previously known lower bounds for TC(X) use the structure of the cohomology algebra of X. In this paper we show how cohomology operations can be used to sharpen these lower bounds for TC(X). As an application of this technique we calculate explicitly the topological complexity of various lens spaces. The results of the paper were inspired by the work of E. Fadell and S. Husseini on weights of cohomology classes appearing in the classical lower bounds for the Lusternik Schnirelmann category. In the appendix to this paper we give a very short proof of a generalized version of their result. 1.
Topological Robotics: Motion Planning
, 2005
"... In this paper we discuss topological problems inspired by robotics. We study in detail the robot motion planning problem. With any pathconnected topological space X we associate a numerical invariant TC(X) measuring the “complexity of the problem of navigation in X. ” We examine how the number TC( ..."
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Cited by 9 (1 self)
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In this paper we discuss topological problems inspired by robotics. We study in detail the robot motion planning problem. With any pathconnected topological space X we associate a numerical invariant TC(X) measuring the “complexity of the problem of navigation in X. ” We examine how the number TC(X) determines the structure of motion planning algorithms, both deterministic and random. We compute the invariant TC(X) in many interesting examples. In the case of the real projective space �P n (where n � 1, 3, 7) the number TC(�P n) − 1 equals the minimal dimension of the Euclidean space into which �P n can be immersed.
Symmetric topological complexity of projective and lens spaces
 ALGEBR. GEOM. TOPOL
, 2009
"... 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 indetermin ..."
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Cited by 8 (3 self)
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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.
Topological complexity of collision free motion planning algorithms in the presence of multiple moving obstacles
, 2006
"... ..."
Presentations of graph braid groups
, 2009
"... Abstract. Let Γ be a graph. The (unlabeled) configuration space UC n Γ of n points on Γ is the space of nelement subsets of Γ. The nstrand braid group of Γ, denoted BnΓ, is the fundamental group of UC n Γ. This paper extends the methods and results of [11]. Here we compute presentations for BnΓ, w ..."
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Cited by 4 (2 self)
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Abstract. Let Γ be a graph. The (unlabeled) configuration space UC n Γ of n points on Γ is the space of nelement subsets of Γ. The nstrand braid group of Γ, denoted BnΓ, is the fundamental group of UC n Γ. This paper extends the methods and results of [11]. Here we compute presentations for BnΓ, where n is an arbitrary natural number and Γ is an arbitrary finite connected graph. Particular attention is paid to the case n = 2, and many examples are given. 1.
On rigidity and the isomorphism problem for tree braid groups
 Groups Geom. Dyn
"... Abstract. We solve the isomorphism problem for braid groups on trees with n = 4 or 5 strands. We do so in three main steps, each of which is interesting in its own right. First, we establish some tools and terminology for dealing with computations using the cohomology of tree braid groups, couching ..."
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Abstract. We solve the isomorphism problem for braid groups on trees with n = 4 or 5 strands. We do so in three main steps, each of which is interesting in its own right. First, we establish some tools and terminology for dealing with computations using the cohomology of tree braid groups, couching our discussion in the language of differential forms. Second, we show that, given a tree braid group BnT on n = 4 or 5 strands, H ∗ (BnT) is an exterior face algebra. Finally, we prove that one may reconstruct the tree T from a tree braid group BnT for n = 4 or 5. Among other corollaries, this third step shows that, when n = 4 or 5, tree braid groups BnT and trees T (up to homeomorphism) are in bijective correspondence. That such a bijection exists is not true for higher dimensional spaces, and is an artifact of the 1dimensionality of trees. We end by stating the results for rightangled Artin groups corresponding to the main theorems, some of which do not yet appear in the literature. 1.