Oberwolfach Preprints (OWP)
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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.
- ItemZ2-Thurston norm and complexity of 3-manifolds, II(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Jaco, William; Rubinstein, J. Hyam; Spreer, Jonathan; Tillmann, StephanIn this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3-manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston's geometry SL2(R)˜.
- ItemGradient canyons, concentration of curvature, and Lipschitz invariants(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Paunescu, Laurentiu; Tibăr, Mihai-MariusWe find new bi-Lipschitz invariants of holomorphic functions of two variables by using the gradient canyons and by combining analytic and geometric viewpoints on the concentration of curvature.
- ItemThe Varchenko determinant of a Coxeter arrangement(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Pfeiffer, Götz; Randriamaro, HeryThe Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relating determinants.
- ItemNon-extendability of holomorphic functions with bounded or continuously extendable derivatives(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Moschonas, Dionysios; Nestoridis, VassiliWe consider the spaces H∞F(Ω) and AF(Ω) containing all holomorphic functions f on an open set Ω⊆C, such that all derivatives f(l), l∈F⊆N0={0,1,...}, are bounded on Ω, or continuously extendable on Ω¯¯¯¯, respectively. We endow these spaces with their natural topologies and they become Fr\'echet spaces. We prove that the set S of non-extendable functions in each of these spaces is either void, or dense and Gδ. We give examples where S=∅ or not. Furthermore, we examine cases where F can be replaced by F˜={l∈N0:minF⩽l⩽supF}, or F˜0={l∈N0:0⩽l⩽supF} and the corresponding spaces stay unchanged.