Topological Complexity of Motion Planning

Abstract. In this paper we study a notion of topological complexity TC (X) for the motion planning problem. TC (X) is a number which measures discontinuity of the process of motion planning in the configuration space X . More precisely, TC (X) is the minimal number k such that there are k different "motion planning rules," each defined on an open subset of X× X , so that each rule is continuous in the source and target configurations. We use methods of algebraic topology (the Lusternik—Schnirelman theory) to study the topological complexity TC (X) . We give an upper bound for TC (X) (in terms of the dimension of the configuration space X ) and also a lower bound (in terms of the structure of the cohomology algebra of X ). We explicitly compute the topological complexity of motion planning for a number of configuration spaces: spheres, two-dimensional surfaces, products of spheres. In particular, we completely calculate the topological complexity of the problem of motion planning for a robot arm in the absence of obstacles.     

On higher analogs of topological complexity

Farber introduced a notion of topological complexity TC(X) that is related to robotics. Here we introduce a series of numerical invariants TCn(X), n=2,3,… , such that TC₂(X)=TC(X) and TCn(X)?TCn+1(X). For these higher complexities, we define their symmetric versions that can also be regarded as higher analogs of the symmetric topological complexity.     

Topological complexity, fibrations and symmetry

We show how locally smooth actions of compact Lie groups on a manifold X can be used to obtain new upper bounds for the topological complexity TC(X), in the sense of Farber. We also obtain new bounds for the topological complexity of finitely generated torsion-free nilpotent groups.     

On the Weight of Nowhere Dense Subsets in Compact Spaces

We study a new cardinal-valued invariant ndw(X) (calling it the nd-weight of X) of a topological space which is defined as the least upper bound of the weights of nowhere dense subsets of X. The main result is the proof of the inequality hl(X)ndw(X) for compact sets without isolated points ((hl is the hereditary Lindelof number). This inequality implies that a compact space without isolated points of countable nd-weight is completely normal. Assuming the continuum hypothesis, we construct an example of a nonmetrizable compact space of countable nd-weight without isolated points.     

On A(X) → TC(X) for some X

Let X be a connected based space and p be a two-regular prime number. If the fundamental group of X has order p, we compute the two-primary homotopy groups of the homotopy fiber of the trace map A(X) → TC(X) relating algebraic K-theory of spaces to topological cyclic homology. The proof uses a theorem of Dundas and an explicit calculation of the cyclotomic trace map K(ℤ[C_p])→ TC(ℤ[C_p]).     

Clone Segments of the Tychonoff Modification of Space

For every natural number n > 1, we construct a topological space X such that the clone segments of its Tychonoff modification T M X verify Clo _k (X) = Clo _k (T M X) if k < n and Clo _n (X) is not isomorphic to Clo _n (T M X).     

PRODUCTS IN TOPOLOGY

Abstract

Examples are provided which demonstrate that in many cases topological products do not behave as they should. A new product for topological spaces is defined in a natural way by means of interior covers. In general this is no longer a topological space but can be interpreted as categorical product in a category larger than Top. For compact spaces the new product coincides with the old. There is a converse: For symmetric topological spaces X the following conditions are equivalent: (1) X is compact; (2) for each cardinal k the old and the new product X^k coincide; (3) for each compact Hausdorff space Y the old and the new product X x Y coincide. The new product preserves paracompactness, zero-dimensionality (in the covering sense), the Lindelöf property, and regular-closedness. With respect to the new product, a space is N-complete iff it is zerodimensional and R-complete.     

Zero-divisor graph of C(X)

Summary In this article the zero-divisor graph Γ(C(X)) of the ring C(X) is studied. We associate the ring properties of C(X), the graph properties of Γ(C(X)) and the topological properties of X. Cycles in Γ(C(X)) are investigated and an algebraic and a topological characterization is given for the graph Γ(C(X)) to be triangulated or hypertriangulated. We have shown that the clique number of Γ(C(X)), the cellularity of X and the Goldie dimension of C(X) coincide. It turns out that the dominating number of Γ(C(X)) is between the density and the weight of X. Finally we have shown that Γ(C(X)) is not triangulated and the set of centers of Γ(C(X)) is a dominating set if and only if the set of isolated points of X is dense in X if and only if the socle of C(X) is an essential ideal.     

Geometrical implications of the existence of very smooth bump functions in Banach spaces

We show that ifX is a Banach space and if there is a non-zero real-valuedC ^∞-smooth function onX with bounded support, then eitherX contains an isomorphic copy ofc ₀(N), or there is an integerk greater than or equal to 1 such thatX is of exact cotype 2k and, in this case,X contains an isomorphic copy ofl ^2k(N). We also show that ifX is a Banach space such that there is onX a non-zero real-valuedC ⁴-smooth function with bounded support and ifX is of cotypeq forq<4, thenX is isomorphic to a Hilbert space.     

Singularities, Expanders and Topology of Maps. Part 1: Homology Versus Volume in the Spaces of Cycles

We find lower bounds on the topological complexity of the critical (values) sets S(F) ì Y{\Sigma(F) \subset Y} of generic smooth maps F : X → Y, as well as on the complexity of the fibers F^-1(y) ì X{F^{-1}(y) \subset X} in terms of the topology of X and Y, where the relevant topological invariants of X are often encoded in the geometry of some Riemannian metric supported by X.     

Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and D _k (X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: D _k (X,Y) is metrizable iff D _k (X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then D _k (X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.     

Hyperbolic rank and subexponential corank of metric spaces

We introduce a new quasi-isometry invariant corank X of a metric space X called subexponential corank. A metric space X has subexponential corank k if roughly speaking there exists a continuous map , T is a topological space, such that for each the set g ^-1(t) has subexponential growth rate in X and the topological dimension dimT = k is minimal among all such maps. Our main result is the inequality for a large class of metric spaces X including all locally compact Hadamard spaces, where rank _h X is the maximal topological dimension of among all CAT(—1) spaces Y quasi-isometrically embedded into X (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of rank _h conjectured by Gromov, in particular, that any Riemannian symmetric space X of noncompact type possesses no quasi-isometric embedding of the standard hyperbolic space H ⁿ with . Submitted: February 2001, Revised: October 2001.     

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Let X be a locally compact topological space and (X, E, X_ω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X_ω ⊆ X, such that all internal subsets of X_ω are relatively compact in the induced topology and X is homeomorphic to the quotient X_ω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function X ? ^*\Bbb CX \rightarrow {}{^{\ast}{\Bbb C}} . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient M_w(X)/M₀(X){\cal M}_{\omega}(X)/{\cal M}_0(X) , for certain external subspaces M₀(X), M_w(X){\cal M}_0(X), {\cal M}_{\omega}(X) of the hyperfinite dimensional Banach space ^*\Bbb C^X{}{^{\ast}{\Bbb C}}^X , with the norm ‖f‖₁ = ∑_{x ∈ X} |f(x)|. If additionally X = G is a hyperfinite group, X_ω = G_ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G₀ of G_ω, and G is isomorphic to the locally compact group G_ω/G₀, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and M_w(G)/M₀(G){\cal M}_{\omega}(G)/{\cal M}_0(G) are isometrically isomorphic as Banach algebras.     

On module bundles and topological K-theory

Abstract  We define and study the group K(X) of a topological space X as the Grothendieck group of the category of suitable module bundles over X instead of the Grothendieck group of the category of vector bundles over X and prove some of its properties. Keywords Topological K-Theory, Module bundles, Waelbroeck algebra Mathematics Subject Classification (2000) 19 Lxx, 19 Axx, (Secondary) 46 Hxx     

Extended equivariant Picard complexes and homogeneous spaces

Let k be a field of characteristic 0 and let [`(k)] \bar{k} be a fixed algebraic closure of k. Let X be a smooth geometrically integral k-variety; we set [`(X)] = X ×_k[`(k)] \bar{X} = X{ \times_k}\bar{k} and denote by [`(X)] \bar{X} . In [BvH2] we defined the extended Picard complex of X as the complex of Gal( [`(k)]

On Sectional Genus of Quasi—Polarized Manifolds with Non-Negative Kodaira Dimension

Let X be a smooth projective variety over ? and L be a nef-big divisor on X. Then (X, L) is called a quasi - polarized manifold. Then we conjecture that g(L) ≥ q(X), where g(L) is the sectional genus of L and q(X) = dim H¹(O_x) is the irregularity of X. In general it is unknown that this conjecture is true or not even in the case of dim X = 2. For example, this conjecture is true if dim X = 2 and dim H≥(L) > 0. But it is unknown if dim X ≥ 3 and dim H⁰(L) > 0. In this paper, we consider a lower bound for g(L) if dim X = 2, dim H⁰(L) ≥ 2, and k(X) ≥ 0. We obtain a stronger result than the above conjecture if dim Bs|L| ≤ 0 by a new method which can be applied to higher dimensional cases. Next we apply this method to the case in which dim X = n ≥ 3 and we obtain a lower bound for g(L) if dim X = 3, dim H⁰(L) ≥ 2, and k(X) ≥ 0.     

Optimally Cutting a Surface into a Disk

We consider the problem of cutting a subset of the edges of a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard in general, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time n ^O(g+k), where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log² g)-approximation of the minimum cut graph in O(g ² n log n) time.     

Duality of <Emphasis Type="Italic">κ</Emphasis>-normed topological vector spaces and their applications

In this paper, the duality of κ-normed topological vector spaces X is defined and investigated, where X is over the field K = R, or K = C, or a non-Archimedean field. For such spaces, an analog of the Mackey-Arens theorem is proved. The conditional κ-normability of spaces L(X) of linear topological homeomorphisms of a locally convex κ-normed space X is studied, where the image of elements under the corresponding operations is in L(X). Cases where the κ-normability of a topological vector space implies its local convexity are investigated. Applications of κ-normed spaces for resolutions of differential equations and for approximations of functions in mathematical economics are given. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 52, Functional Analysis, 2008.