scholarly journals Combinatorics of Orbit Configuration Spaces

2020 ◽  
Author(s):  
Christin Bibby ◽  
Nir Gadish
Keyword(s):  
Configuration Space ◽  
Group Action ◽  
Cartesian Product ◽  
Independent Interest ◽  
Configuration Spaces ◽  
Connected Components ◽  
Combinatorial Description ◽  
Close Relationship ◽  
Dowling Lattices

Abstract From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this “orbit configuration space” is the complement of an arrangement of subvarieties inside the Cartesian product, and we use this structure to study its topology. We give an abstract combinatorial description of its poset of layers (connected components of intersections from the arrangement), which turns out to be of much independent interest as a generalization of partition and Dowling lattices. The close relationship to these classical posets is then exploited to give explicit cohomological calculations.

scholarly journals Cohomology of generalized configuration spaces

2019 ◽  
Vol 156 (2) ◽  
pp. 251-298
Author(s):  
Dan Petersen
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Configuration Space ◽  
Compact Support ◽  
Cartesian Product ◽  
Configuration Spaces ◽  
Dg Algebra ◽  
Systematic Framework

Let $X$ be a topological space. We consider certain generalized configuration spaces of points on $X$, obtained from the cartesian product $X^{n}$ by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call ‘twisted commutative dg algebra models’ for the cochains on $X$. Suppose that $X$ is a ‘nice’ topological space, $R$ is any commutative ring, $H_{c}^{\bullet }(X,R)\rightarrow H^{\bullet }(X,R)$ is the zero map, and that $H_{c}^{\bullet }(X,R)$ is a projective $R$-module. We prove that the compact support cohomology of any generalized configuration space of points on $X$ depends only on the graded $R$-module $H_{c}^{\bullet }(X,R)$. This generalizes a theorem of Arabia.

scholarly journals Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Prashanth Raman ◽  
Chi Zhang
Keyword(s):  
Large Class ◽  
Configuration Space ◽  
Moduli Space ◽  
Cluster Algebras ◽  
Configuration Spaces ◽  
Minkowski Sum ◽  
Canonical Forms ◽  
Infinite Class ◽  
Special Cases ◽  
Generalized Associahedra

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

scholarly journals Regions of Feasible Point-to-Point Trajectories in the Cartesian Workspace of Fully-Parallel Manipulators

1999 ◽  
Author(s):  
Damien Chablat ◽  
Philippe Wenger
Keyword(s):  
Parallel Manipulator ◽  
Cartesian Product ◽  
Parallel Manipulators ◽  
Joint Space ◽  
Inverse Kinematic ◽  
Connected Components ◽  
Feasible Point ◽  
Cartesian Space ◽  
Point Trajectories ◽  
Point To Point

Abstract The goal of this paper is to define the n-connected regions in the Cartesian workspace of fully-parallel manipulators, i.e. the maximal regions where it is possible to execute point-to-point motions. The manipulators considered in this study may have multiple direct and inverse kinematic solutions. The N-connected regions are characterized by projection, onto the Cartesian workspace, of the connected components of the reachable configuration space defined in the Cartesian product of the Cartesian space by the joint space. Generalized octree models are used for the construction of all spaces. This study is illustrated with a simple planar fully-parallel manipulator.

scholarly journals On the Cohomology of the Mapping Class Group of the Punctured Projective Plane

2020 ◽  
Vol 71 (2) ◽  
pp. 539-555
Author(s):  
Miguel A Maldonado ◽  
Miguel A Xicoténcatl
Keyword(s):  
Exact Sequence ◽  
Configuration Space ◽  
Mapping Class Group ◽  
Homotopy Theory ◽  
Class Group ◽  
Orientable Surface ◽  
Configuration Spaces ◽  
Mapping Class ◽  
Serre Spectral Sequence ◽  
Classical Homotopy

Abstract The mapping class group $\Gamma ^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of ${\mathbb{R}} \textrm{P}^2$, we analyze the Serre spectral sequence of a fiber bundle $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k \to X_k \to BSO(3)$ where $X_k$ is a $K(\Gamma ^k({\mathbb{R}} \textrm{P}^2),1)$ and $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$ denotes the configuration space of unordered $k$-tuples of distinct points in ${\mathbb{R}} \textrm{P}^2$. As a consequence, we express the mod-2 cohomology of $\Gamma ^k({\mathbb{R}} \textrm{P}^2)$ in terms of that of $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$.

scholarly journals Free space of rigid objects: caging, path non-existence, and narrow passage detection

2020 ◽  
pp. 027836492093299
Author(s):  
Anastasiia Varava ◽  
J. Frederico Carvalho ◽  
Danica Kragic ◽  
Florian T. Pokorny
Keyword(s):  
Configuration Space ◽  
Three Dimensions ◽  
Robotic Manipulation ◽  
Configuration Spaces ◽  
Finite Sample ◽  
Fine Grained ◽  
Rigid Objects ◽  
Uniform Grids ◽  
Narrow Passage ◽  
Object Shapes

In this work, we propose algorithms to explicitly construct a conservative estimate of the configuration spaces of rigid objects in two and three dimensions. Our approach is able to detect compact path components and narrow passages in configuration space which are important for applications in robotic manipulation and path planning. Moreover, as we demonstrate, they are also applicable to identification of molecular cages in chemistry. Our algorithms are based on a decomposition of the resulting three- and six-dimensional configuration spaces into slices corresponding to a finite sample of fixed orientations in configuration space. We utilize dual diagrams of unions of balls and uniform grids of orientations to approximate the configuration space. Furthermore, we carry out experiments to evaluate the computational efficiency on a set of objects with different geometric features thus demonstrating that our approach is applicable to different object shapes. We investigate the performance of our algorithm by computing increasingly fine-grained approximations of the object’s configuration space. A multithreaded implementation of our approach is shown to result in significant speed improvements.

HARMONIC ANALYSIS ON CONFIGURATION SPACE I: GENERAL THEORY

2002 ◽  
Vol 05 (02) ◽  
pp. 201-233 ◽  
Author(s):  
YURI G. KONDRATIEV ◽  
TOBIAS KUNA
Keyword(s):  
Fourier Transform ◽  
Harmonic Analysis ◽  
Configuration Space ◽  
Riemannian Manifolds ◽  
Characterization Theorem ◽  
Configuration Spaces ◽  
Probability Measures ◽  
Correlation Measures ◽  
Underlying Manifold ◽  
Lifting Operator

We develop a combinatorial version of harmonic analysis on configuration spaces over Riemannian manifolds. Our constructions are based on the use of a lifting operator which can be considered as a kind of (combinatorial) Fourier transform in the configuration space analysis. The latter operator gives us a natural lifting of the geometry from the underlying manifold onto the configuration space. Properties of correlation measures for given states (i.e. probability measures) on configuration spaces are studied including a characterization theorem for correlation measures.

scholarly journals MORSE THEORY AND THE TOPOLOGY OF CONFIGURATION SPACE

1996 ◽  
Vol 11 (05) ◽  
pp. 823-843
Author(s):  
W.D. McGLINN ◽  
L. O’RAIFEARTAIGH ◽  
S. SEN ◽  
R.D. SORKIN
Keyword(s):  
Potential Function ◽  
Configuration Space ◽  
Morse Theory ◽  
Three Dimensional ◽  
Configuration Spaces ◽  
Arbitrary Field ◽  
Two Dimensional ◽  
Point Particles ◽  
Homology Groups ◽  
Fermi Statistics

The first and second homology groups, H1 and H2, are computed for configuration spaces of framed three-dimensional point particles with annihilation included, when up to two particles and an antiparticle are present, the types of frames considered being S2 and SO(3). Whereas a recent calculation for two-dimensional particles used the Mayer–Vietoris sequence, in the present work Morse theory is used. By constructing a potential function none of whose critical indices is less than four, we find that (for coefficients in an arbitrary field K) the homology groups H1 and H2 reduce to those of the frame space, S2 or SO(3) as the case may be. In the case of SO(3) frames this result implies that H1 (with coefficients in ℤ2) is generated by the cycle corresponding to a 2π rotation of the frame. (This same cycle is homologous to the exchange loop: the spin-statistics correlation.) It also implies that H2 is trivial, which means that there does not exist a topologically nontrivial Wess–Zumino term for SO(3) frames [in contrast to the two-dimensional case, where SO(2) frames do possess such a term]. In the case of S2 frames (with coefficients in ℝ), we conclude H2=ℝ, the generator being in effect the frame space itself. This implies that for S2 frames there does exist a Wess–Zumino term, as indeed is needed for the possibility of half-integer spin and the corresponding Fermi statistics. Taken together, these results for H1 and H2 imply that our configuration space “admits spin 1/2” for either choice of frame, meaning that the spin-statistics theorem previously proved for this space is not vacuous.

scholarly journals Topological complexity of collision-free motion planning on surfaces

2010 ◽  
Vol 147 (2) ◽  
pp. 649-660 ◽  
Author(s):  
Daniel C. Cohen ◽  
Michael Farber
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Main Tool ◽  
Orientable Surface ◽  
Configuration Spaces ◽  
Topological Complexity ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Lusternik Schnirelmann ◽  
Mixed Hodge Structures

AbstractThe topological complexity$\mathsf {TC}(X)$is a numerical homotopy invariant of a topological spaceXwhich is motivated by robotics and is similar in spirit to the classical Lusternik–Schnirelmann category ofX. Given a mechanical system with configuration spaceX, the invariant$\mathsf {TC}(X)$measures the complexity of motion planning algorithms which can be designed for the system. In this paper, we compute the topological complexity of the configuration space ofndistinct ordered points on an orientable surface, for both closed and punctured surfaces. Our main tool is a theorem of B. Totaro describing the cohomology of configuration spaces of algebraic varieties. For configuration spaces of punctured surfaces, this is used in conjunction with techniques from the theory of mixed Hodge structures.

scholarly journals The configuration space of a robotic arm in a tunnel of width 2

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Federico Ardila ◽  
Hanner Bastidas ◽  
Cesar Ceballos ◽  
John Guo
Keyword(s):  
Configuration Space ◽  
Robotic Arm ◽  
Configuration Spaces ◽  
Cubical Complex ◽  
International Audience ◽  
Rectangular Tunnel

International audience We study the motion of a robotic arm inside a rectangular tunnel of width 2. We prove that the configuration space S of all possible positions of the robot is a CAT(0) cubical complex. Before this work, very few families of robots were known to have CAT(0) configuration spaces. This property allows us to move the arm optimally from one position to another.

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