Searcharxiv⌕ Search

arXiv subjects

Michael Farber

Publications and source records attributed to Michael Farber.

At least 73 records · Page 4Linked to original sources
arXiv2006

Cohomological estimates for $\cat(X,ξ)$

This paper studies the homotopy invariant $\cat(X,ξ)$ introduced in \cite{farbe2}. Given a finite cell-complex $X$, we study the function $ξ\mapsto \cat(X,ξ)$ where $ξ$ varies in the cohomology space $H^1(X;\R)$. Note that $\cat(X,ξ)$ turns into the classical Lusternik - Schnirelmann category $\cat(X)$ in the case $ξ=0$. Interest in $\cat(X,ξ)$ is based on its applications in dynamics where it enters estimates of complexity of the chain recurrent set of a flow admitting Lyapunov closed 1-forms.

math.AT↗
arXiv2004

Collision Free Motion Planning on Graphs

A topological theory initiated recently by the author uses methods of algebraic topology to estimate numerically the character of instabilities arising in motion planning algorithms. The present paper studies random motion planning algorithms and reveals how the topology of the robot's configuration space influences their structure. We prove that the topological complexity of motion planning TC(X) coincides with the minimal n such that there exists an n-valued random motion planning algorithm for the system; here $X$ denotes the configuration space. We study in detail the problem of collision free motion of several objects on a graph G. We describe an explicit motion planning algorithm for this problem. We prove that if G is a tree and if the number of objects is large enough, then the topological complexity of this motion planning problem equals 2m(G)+1 where m(G) is the number of the essential vertices of G. It turns out (in contrast with the results on the collision free control of many objects in space obtained earlier jointly with S. Yuzvinsky) that the topological complexity is independent of the number of particles.

math.AT↗
arXiv2002

Topological Robotics: Subspace Arrangements and Collision Free Motion Planning

We study an elementary problem of the topological robotics: collective motion of a set of $n$ distinct particles which one has to move from an initial configuration to a final configuration, with the requirement that no collisions occur in the process of motion. The ultimate goal is to construct an algorithm which will perform this task once the initial and the final configurations are given. This reduces to a topological problem of finding the topological complexity TC(C_n(\R^m)) of the configutation space C_n(\R^m) of $n$ distinct ordered particles in \R^m. We solve this problem for m=2 (the planar case) and for all odd m, including the case m=3 (particles in the three-dimensional space). We also study a more general motion planning problem in Euclidean space with a hyperplane arrangement as obstacle.

math.AT↗
arXiv2002

Topological robotics: motion planning in projective spaces

We study an elementary problem of topological robotics: rotation of a line, which is fixed by a revolving joint at a base point: one wants to bring the line from its initial position to a final position by a continuous motion in the space. The final goal is to construct an algorithm which will perform this task once the initial and final positions are given. Any such motion planning algorithm will have instabilities, which are caused by topological reasons. A general approach to study instabilities of robot motion was suggested recently by the first named author. With any path-connected topological space X one associates a number TC(X), called the topological complexity of X. This number is of fundamental importance for the motion planning problem: TC(X) determines character of instabilities which have all motion planning algorithms in X. In the present paper we study the topological complexity of real projective spaces. In particular we compute TC(RP^n) for all n<24. Our main result is that (for n distinct from 1, 3, 7) the problem of calculating of TC(RP^n) is equivalent to finding the smallest k such that RP^n can be immersed into the Euclidean space R^{k-1}.

math.AT↗
arXiv2002

Instabilities of Robot Motion

Instabilities of robot motion are caused by topological reasons. In this paper we find a relation between the topological properties of a configuration space (the structure of its cohomology algebra) and the character of instabilities, which are unavoidable in any motion planning algorithm. More specifically, let $X$ denote the space of all admissible configurations of a mechanical system. A {\it motion planner} is given by a splitting $X\times X = F_1\cup F_2\cup ... \cup F_k$ (where $F_1, ..., F_k$ are pairwise disjoint ENRs, see below) and by continuous maps $s_j: F_j \to PX,$ such that $E\circ s_j =1_{F_j}$. Here $PX$ denotes the space of all continuous paths in $X$ (admissible motions of the system) and $E: PX\to X\times X$ denotes the map which assigns to a path the pair of its initial -- end points. Any motion planner determines an algorithm of motion planning for the system. In this paper we apply methods of algebraic topology to study the minimal number of sets $F_j$ in any motion planner in $X$. We also introduce a new notion of {\it order of instability} of a motion planner; it describes the number of essentially distinct motions which may occur as a result of small perturbations of the input data. We find the minimal order of instability, which may have motion planners on a given configuration space $X$. We study a number of specific problems: motion of a rigid body in $\R^3$, a robot arm, motion in $\R^3$ in the presence of obstacles, and others.

cs.RO↗
arXiv2002

Lusternik - Schnirelman theory and dynamics

In this paper we study a new topological invariant $\Cat(X,ξ)$, where $X$ is a finite polyhedron and $ξ\in H^1(X;\R)$ is a real cohomology class. $\Cat(X,ξ)$ is defined using open covers of $X$ with certain geometric properties; it is a generalization of the classical Lusternik -- Schnirelman category. We show that $\Cat(X,ξ)$ depends only on the homotopy type of $(X,ξ)$. We prove that $\Cat(X,ξ)$ allows to establish a relation between the number of equilibrium states of dynamical systems and their global dynamical properties (such as existence of homoclinic cycles and the structure of the set of chain recurrent points). In the paper we give a cohomological lower bound for $\Cat(X,ξ)$, which uses cup-products of cohomology classes of flat line bundles with monodromy described by complex numbers, which are not Dirichlet units.

math.AT↗
arXiv2001

Topological complexity of motion planning

In this paper we study a notion of topological complexity for the motion planning problem. The topological complexity is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely, it is the minimal number k such that there are k different motion planning rules, each defined on an open subset of XxX, 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 . We give an upper bound (in terms of the dimension of the configuration space X) and also a lower bound (in the 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: for spheres, two-dimensional surfaces, for 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.

math.AT↗
arXiv2001

Zeros of closed 1-forms, homoclinic orbits, and Lusternik - Schnirelman theory

In this paper we study topological lower bounds on the number of zeros of closed 1-forms without Morse type assumptions. We prove that one may always find a representing closed 1-form having at most one zero. We introduce and study a generalization $cat(X,ξ)$ of the notion of Lusternik - Schnirelman category, depending on a topological space $X$ and a cohomology class $ξ\in H^1(X;\R)$. We prove that any closed 1-form has at least $cat(X,ξ)$ zeros assuming that it admits a gradient-like vector field with no homoclinic cycles. We show that the number $cat(X,ξ)$ can be estimated from below in terms of the cup-products and higher Massey products. This paper corrects some statements made in my previous papers on this subject.

math.DG↗
arXiv2001

Periodic trajectories in 3-dimensional convex billiards

We give lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n-1) such trajectories. We apply a topological approach based on the calculation of cohomology of certain configuration spaces.

math.DG↗
arXiv2000

Novikov - Shubin signatures, II

This paper continues math.DG/9903140. Here we construct a linking form on the torsion part of middle dimensional extended L^2 homology and cohomology of odd-dimensional manifolds. We give a geometric necessary condition when this linking form is hyperbolic. We compute this linking form in case when the manifold bounds. We introduce and study new numerical invariants of the linking form: the Novikov - Shubin signature and the torsion signature. We compute these invariants explicitly for manifolds with $π_1 = Z$ in terms of the Blanchfield form. We develop a notion of excess for extensions of torsion modules and show how this concept can be used to guarantee vanishing of the torsion signature.

math.DG↗
arXiv2000

Topology of billiard problems, II

We give topological lower bounds on the number of periodic and closed trajectories in strictly convex smooth billiards. We use variational reduction admitting a finite group of symmetries and apply topological approach based on equivariant Morse and Lusternik - Schnirelman theories. The paper continues results published in math.DG/9911226 and math.DG/0006049

math.AT↗
arXiv1999

Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards

We give lowed bounds on the number of periodic trajectories in strictly convex smooth billiards in $\R^{m+1}$ for $m\ge 3$. For plane billiards (when m=1) such bounds were obtained by G. Birkhoff in the 1920's. Our proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik - Schirelman theories. We compute the equivariant cohomology ring of the cyclic configuration space of the sphere $S^m$, i.e., the space of n-tuples of points $(x_1, ..., x_n)$, where $x_i\in S^m$ and $x_i\ne x_{i+1}$ for i=1,2, ..., n.

math.DG↗
arXiv1999

Counting zeros of closed 1-forms

The paper suggests new topological lower bounds for the number of zeros of closed 1-forms within a given cohomology class. The main new technical tool is the deformation complex, which allows to pass to a singular limit and reduce the original problem with a closed 1-form to a traditional problem with a Morse function. We show by examples that the suggested approach may provide stronger estimates than the Novikov inequalities. The technique of the paper also applies to study topology of the set of zeros of closed 1-forms under Bott non-degeneracy assumptions.

math.DG↗
arXiv1999

Novikov - Shubin signatures, I

Torsion objects of von Neumann categories describe the phenomen "spectrum near zero" discovered by S. Novikov and M. Shubin. In this paper we classify Hermitian forms on torsion objects of a finite von Neumann category. We prove that any such form can be represented as a discriminant form of a degenerate Hermitian form on a projective module. We also find a relation between the Hermitian forms on projective modules which holds if and only if their discriminant forms are congruent. A notion of superfinite von Neumann category is introduced. It is proven that the classification of torsion Hermitian forms in a superfinite category can be completely reduced to the isomorphisn types of their positive and negative parts.

math.DG↗
arXiv1999

Absolute torsion and eta-invariant

In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion it has the advantage of having a well-defined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister torsion. In this paper I show that the sign behavior of the absolute torsion, under a continuous deformation of the flat bundle, is determined by the eta-invariant and the Pontrjagin classes.

math.DG↗
arXiv1998

The Morse-Novikov theory of circle-valued functions and noncommutative localization

We use noncommutative localization to construct a chain complex which counts the critical points of a circle-valued Morse function on a manifold, generalizing the Novikov complex. As a consequence we obtain new topological lower bounds on the minimum number of critical points of a circle-valued Morse function within a homotopy class, generalizing the Novikov inequalities.

math.DG↗