Browsing by Author "Lupton, Gregory"
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- ItemBredon cohomology and robot motion planning(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Farber, Michael; Grant, Mark; Lupton, Gregory; Oprea, JohnIn this paper we study the topological invariant TC(X) reflecting the complexity of algorithms for autonomous robot motion. Here, X stands for the configuration space of a system and TC(X) is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in X. We focus on the case when the space X is aspherical; then the number TC(X) depends only on the fundamental group π=π1(X) and we denote it TC(π). We prove that TC(π) can be characterised as the smallest integer k such that the canonical π×π-equivariant map of classifying spaces E(π×π)→ED(π×π) can be equivariantly deformed into the k-dimensional skeleton of ED(π×π). The symbol E(π×π) denotes the classifying space for free actions and ED(πtimesπ) denotes the classifying space for actions with isotropy in a certain family D of subgroups of π×π. Using this result we show how one can estimate TC(π) in terms of the equivariant Bredon cohomology theory. We prove that TC(π)≤max{3,cdD(π×π)}, where cdD(π×π) denotes the cohomological dimension of π×π with respect to the family of subgroups D. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family D.
- ItemHomotopy Theory of Function Spaces and Related Topics(Zürich : EMS Publ. House, 2009) Lupton, Gregory; Smith, SamuelThis workshop brought together researchers studying a variety of problems related to the homotopy theory of function spaces. Topics covered included: evaluation maps and Gottlieb groups, the classification of gauge groups and of other function space components, algebraic models for function spaces both in the rational and in the p-local settings, operads, configuration spaces, free and based loop spaces and infinite-dimensional Lie groups.
- ItemMini-Workshop: Topological Complexity and Related Topics(Zürich : EMS Publ. House, 2016) Lupton, Gregory; Vandembroucq, LucileTopological complexity is a numerical homotopy invariant of topological spaces, of Lusternik-Schnirelmann type, introduced by Farber and motivated by the motion planning problem from topological robotics. This mini-workshop assembled researchers interested in calculating the topological complexity and its many variants, with the aim of providing a snapshot of the current state of knowledge, and shaping directions of future research.
- ItemRealizing spaces as classifying spaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2015) Lupton, Gregory; Smith, Samuel BruceWhich spaces occur as a classifying space for fibrations with a given fibre? We address this question in the context of rational homotopy theory. We construct an infinite family of finite complexes realized (up to rational homotopy) as classifying spaces. We also give several non-realization results, including the following: the rational homotopy types of CP2 and S4 are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.
- ItemTopological Complexity, Robotics and Social Choice(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Carrasquel, José; Lupton, Gregory; Oprea, JohnTopological complexity is a number that measures how hard it is to plan motions (for robots, say) in terms of a particular space associated to the kind of motion to be planned. This is a burgeoning subject within the wider area of Applied Algebraic Topology. Surprisingly, the same mathematics gives insight into the question of creating social choice functions, which may be viewed as algorithms for making decisions by artificial intelligences.