scholarly journals On generalized configuration space and its homotopy groups

2020 ◽  
pp. 2043001
Author(s):  
Jun Wang ◽  
Xuezhi Zhao
Keyword(s):  
Projective Space ◽  
Configuration Space ◽  
Topological Property ◽  
Configuration Spaces ◽  
Stiefel Manifold ◽  
Homotopy Groups ◽  
Fundamental Groups ◽  
Stiefel Manifolds ◽  
Linearly Independent ◽  
Special Cases

Let [Formula: see text] be a subset of vector space or projective space. The authors define generalized configuration space of [Formula: see text] which is formed by [Formula: see text]-tuples of elements of [Formula: see text], where any [Formula: see text] elements of each [Formula: see text]-tuple are linearly independent. The generalized configuration space gives a generalization of Fadell’s classical configuration space, and Stiefel manifold. Denote generalized configuration space of [Formula: see text] by [Formula: see text]. For studying topological property of the generalized configuration spaces, the authors calculate homotopy groups for some special cases. This paper gives the fundamental groups of generalized configuration spaces of [Formula: see text] for some special cases, and the connections between the homotopy groups of generalized configuration spaces of [Formula: see text] and the homotopy groups of Stiefel manifolds. It is also proved that the higher homotopy groups of generalized configuration spaces [Formula: see text] and [Formula: see text] are isomorphic.

scholarly journals Whitehead products in the complex Stiefel manifolds

1983 ◽  
Vol 26 (2) ◽  
pp. 241-251 ◽  
Author(s):  
Yasukuni Furukawa
Keyword(s):  
Stiefel Manifold ◽  
Isotropy Group ◽  
Homotopy Groups ◽  
Stiefel Manifolds

The complex Stiefel manifoldWn,k, wheren≦k≦1, is a space whose points arek-frames inCn. By using the formula of McCarty [4], we will make the calculations of the Whitehead products in the groups π*(Wn,k). The case of real and quaternionic will be treated by Nomura and Furukawa [7]. The product [[η],j1l] appears as generator of the isotropy group of the identity map of Stiefel manifolds. In this note we use freely the results of the 2-components of the homotopy groups of real and complex Stiefel manifolds such as Paechter [8], Hoo-Mahowald [1], Nomura [5], Sigrist [9] and Nomura-Furukawa [6].

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.

On complex Stiefel manifolds

1960 ◽  
Vol 56 (4) ◽  
pp. 342-353 ◽  
Author(s):  
M. F. Atiyah ◽  
J. A. Todd
Keyword(s):  
Projective Space ◽  
Unitary Group ◽  
Complex Projective Space ◽  
Natural Fibre ◽  
Complex Case ◽  
Stiefel Manifold ◽  
Dimensional Sphere ◽  
Stiefel Manifolds ◽  
Complex Projective

In a recent series of papers (10), (11), (12), I. M. James has made an illuminating study of Stiefel manifolds. We shall begin by describing his results (for the complex case). Let Wn, k, for k > 1, denote the complex Stiefel manifold U(n)/U(n − k), where U(n) is the unitary group in n variables. Then we have a natural fibre map Wn, k → Wn, 1 = S2n−1, where Sr denotes the r-dimensional sphere. Let Pn, k, for k ≥ 1, denote the ‘stunted complex projective space’ obtained from the (n − 1)-dimensional complex projective space† Pn by identifying to a point a subspace Pn−k. Then we have a natural ‘cofibre map’ Pn, k → Pn, 1 = S2n−2. The space Pn, k is said to be S-reducible if some suspension of the map Pn, k → S2n−2 has a right homotopy inverse. The results of James can then be summarized as follows.

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.

Properties of Selick's filtration of the double suspension E2

2014 ◽  
Vol 06 (03) ◽  
pp. 421-440
Author(s):  
Jelena Grbić ◽  
Stephen Theriault ◽  
Hao Zhao
Keyword(s):  
Hopf Algebra ◽  
Spectral Sequence ◽  
Homotopy Groups ◽  
Classifying Space ◽  
Special Cases ◽  
Text Study ◽  
Classical Homotopy

To help study the double suspension [Formula: see text] when localised at a prime p, Selick filtered Ω2S2n+1 by H-spaces which geometrically realise a natural Hopf algebra filtration of H*(Ω2S2n+1;ℤ/p). Later, Gray showed that the fiber Wn of E2 has an integral classifying space BWn and there is a homotopy fibration [Formula: see text]. In this paper we correspondingly filter BWn in a manner compatible with Selick's filtration and the homotopy fibration [Formula: see text], study the multiplicative properties and homotopy exponents of the spaces in the filtrations, and use the filtrations to filter exponent information for the homotopy groups of S2n+1. Our results link three seemingly different in nature classical homotopy fibrations given by Toda, Selick and Gray and make them special cases of a systematic whole. In addition we construct a spectral sequence which converges to the homotopy groups of BWn.

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.

Osculatory properties of a certain curve in [n]

1974 ◽  
Vol 75 (3) ◽  
pp. 331-344 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Projective Space ◽  
Homogeneous Coordinates ◽  
Linearly Independent ◽  
Image Position

1. When, as will be presumed henceforward, no two of a0, a1, …, an are equal the n + 1 equationsare linearly independent; x0, x1, …, xn are homogeneous coordinates in [n] projective space of n dimensions—and the simplex of reference S is self-polar for all the quadrics.

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.

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