arXiv subjects

Michael Farber

Publications and source records attributed to Michael Farber.

At least 37 records · Page 2Linked to original sources

arXiv2020

Ample simplicial complexes

Motivated by potential applications in network theory, engineering and computer science, we study $r$-ample simplicial complexes. These complexes can be viewed as finite approximations to the Rado complex which has a remarkable property of {\it indestructibility,} in the sense that removing any finite number of its simplexes leaves a complex isomorphic to itself. We prove that an $r$-ample simplicial complex is simply connected and $2$-connected for $r$ large. The number $n$ of vertexes of an $r$-ample simplicial complex satisfies $\exp(Ω(\frac{2^r}{\sqrt{r}}))$. We use the probabilistic method to establish the existence of $r$-ample simplicial complexes with $n$ vertexes for any $n>r 2^r 2^{2^r}$. Finally, we introduce the iterated Paley simplicial complexes, which are explicitly constructed $r$-ample simplicial complexes with nearly optimal number of vertexes.

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arXiv2020

Generating functions and topological complexity

We examine the rationality conjecture which states that (a) the formal power series $\sum_{r\ge 1} \tc_{r+1}(X)\cdot x^r$ represents a rational function of $x$ with a single pole of order 2 at $x=1$ and (b) the leading coefficient of the pole equals $\cat(X)$. Here $X$ is a finite CW-complex and for $r\ge 2$ the symbol $\tc_r(X)$ denotes its $r$-th sequential topological complexity. We analyse an example (violating the Ganea conjecture) and conclude that part (b) of the rationality conjecture is false in general. Besides, we establish a cohomological version of the rationality conjecture.

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arXiv2020

The Rado Simplicial Complex

A Rado simplicial complex X is a generalisation of the well-known Rado graph. X is a countable simplicial complex which contains any countable simplicial complex as its induced subcomplex. The Rado simplicial complex is highly symmetric, it is homogeneous: any isomorphism between finite induced subcomplexes can be extended to an isomorphism of the whole complex. We show that the Rado complex X is unique up to isomorphism and suggest several explicit constructions. We also show that a random simplicial complex on countably many vertices is a Rado complex with probability 1. The geometric realisation |X| of a Rado complex is contractible and is homeomorphic to an infinite dimensional simplex. We also prove several other interesting properties of the Rado complex X, for example we show that removing any finite set of simplexes of X gives a complex isomorphic to X.

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arXiv2020

Topological embeddings into random 2-complexes

We consider 2-dimensional random simplicial complexes $Y$ in the multi-parameter model. We establish the multi-parameter threshold for the property that every 2-dimensional simplicial complex $S$ admits a topological embedding into $Y$ asymptotically almost surely. Namely, if in the procedure of the multi-parameter model, each $i$-dimensional simplex is taken independently with probability $p_i=p_i(n)$, from a set of $n$ vertices, then the threshold is $p_0 p_1^3 p_2^2 = \frac{1}{n}$. This threshold happens to coincide with the previously established thresholds for uniform hyperbolicity and triviality of the fundamental group. Our claim in one direction is in fact slightly stronger, namely, we show that if $p_0 p_1^3 p_2^2$ is sufficiently larger than $\frac{1}{n}$ then every $S$ has a fixed subdivision $S'$ which admits a simplicial embedding into $Y$ asymptotically almost surely. The main geometric result we prove to this end is that given $ε>0$, there is a subdivision $S'$ of $S$ such that every subcomplex $T \subseteq S'$ has $\frac{f_0(T)}{f_1(T)}>\frac{1}{3}-ε$ and $\frac{f_0(T)}{f_2(T)}>\frac{1}{2}-ε$, where $f_i(T)$ denotes the number of simplices in $T$ of dimension $i$. In the other direction we show that if $p_0 p_1^3 p_2^2$ is sufficiently smaller than $\frac{1}{n}$, then asymptotically almost surely, the torus does not admit a topological embedding into $Y$. Here we use a result of Z. Gao which bounds the number of different triangulations of a surface.

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arXiv2019

Random Simplicial Complexes in the Medial Regime

We describe topology of random simplicial complexes in the lower and upper models in the medial regime, i.e. under the assumption that the probability parameters $p_σ$ approach neither $0$ nor $1$. We show that nontrivial Betti numbers of typical lower and upper random simplicial complexes in the medial regime lie in a narrow range of dimensions. For instance, an upper random simplicial complex $Y$ on $n$ vertices in the medial regime with high probability has non-vanishing Betti numbers $b_{j}(Y)$ only for $k+c <n-j<k+\log_2 k +c'$ where $k=\log_2 \ln n$ and $c, c' $ are constants. A lower random simplicial complex on $n$ vertices in the medial regime is with high probability $(k+a)$-connected and its dimension $d$ satisfies $d\sim k+\log_2 k+ a'$ where $a, \, a'$ are constants. The paper develops a new technique, based on Alexander duality, which relates the lower and upper models.

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arXiv2019

Higher topological complexity of aspherical spaces

In this article we study the higher topological complexity ${\sf TC}_r(X)$ in the case when $X$ is an aspherical space, $X=K(π, 1)$ and $r\ge 2$. We give a characterisation of ${\sf TC}_r(K(π, 1))$ in terms of classifying spaces for equivariant Bredon cohomology. Our recent paper \cite{FGLO}, joint with M. Grant and G. Lupton, treats the special case $r=2$. We also obtain in this paper useful lower bounds for ${\sf TC}_r(π)$ in terms of cohomological dimension of subgroups of $π\timesπ\times \dots\times π$ ($r$ times) with certain properties. As an illustration of the main technique we find the higher topological complexity of the Higman's groups. We also apply our method to obtain a lower bound for the higher topological complexity of the right angled Artin (RAA) groups, which, as was established in \cite{GGY} by a different method (in a more general situation), coincides with the precise value. We finish the paper by a discussion of the ${\sf TC}$-generating function $\sum_{r=1}^\infty {\sf TC}_{r+1}(X)x^r$ encoding the values of the higher topological complexity ${\sf TC}_r(X)$ for all values of $r$. We show that in many examples (including the case when $X=K(H, 1)$ with $H$ being a RAA group) the ${\sf TC}$-generating function is a rational function of the form $\frac{P(x)}{(1-x)^2}$ where $P(x)$ is an integer polynomial with $P(1)={\sf cat}(X)$.

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arXiv2018

Directed topological complexity

It has been observed that the very important motion planning problem of robotics mathematically speaking boils down to the problem of finding a section to the path-space fibration, raising the notion of topological complexity, as introduced by M. Farber. The above notion fits the motion planning problem of robotics when there are no constraints on the actual control that can be applied to the physical apparatus. In many applications, however, a physical apparatus may have constrained controls, leading to constraints on its potential future dynamics. In this paper we adapt the notion of topological complexity to the case of directed topological spaces, which encompass such controlled systems, and also systems which appear in concurrency theory. We study its first properties, make calculations for some interesting classes of spaces, and show applications to a form of directed homotopy equivalence.

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arXiv2018

Bredon cohomology and robot motion planning

In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\sf {TC}}(X)$ 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 ${\sf TC}(X)$ depends only on the fundamental group $π=π_1(X)$ and we denote it ${\sf TC}(π)$. We prove that ${\sf TC}(π)$ can be characterised as the smallest integer $k$ such that the canonical $π\timesπ$-equivariant map of classifying spaces $$E(π\timesπ) \to E_{\mathcal D}(π\timesπ)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal D}(π\timesπ)$. The symbol $E(π\timesπ)$ denotes the classifying space for free actions and $E_{\mathcal D}(π\timesπ)$ denotes the classifying space for actions with isotropy in a certain family $\mathcal D$ of subgroups of $π\timesπ$. Using this result we show how one can estimate ${\sf TC}(π)$ in terms of the equivariant Bredon cohomology theory. We prove that ${\sf TC}(π) \le \max\{3, {\rm cd}_{\mathcal D}(π\timesπ)\},$ where ${\rm cd}_{\mathcal D}(π\timesπ)$ denotes the cohomological dimension of $π\timesπ$ with respect to the family of subgroups $\mathcal D$. 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 $\mathcal D$.

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arXiv2018

An upper bound for topological complexity

In arXiv:1711.10132 a new approximating invariant ${\mathsf{TC}}^{\mathcal{D}}$ for topological complexity was introduced called $\mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of ${\mathsf{TC}}^{\mathcal{D}}$ and the connections between ${\mathsf{TC}}^{\mathcal{D}}$ and invariants of Lusternik-Schnirelmann type. We also introduce a new $\mathsf{TC}$-type invariant $\widetilde{\mathsf{TC}}$ that can be used to give an upper bound for $\mathsf{TC}$, $$\mathsf{TC}(X)\le {\mathsf{TC}}^{\mathcal{D}}(X) + \left\lceil \frac{2\dim X -k}{k+1}\right\rceil,$$ where $X$ is a finite dimensional simplicial complex with $k$-connected universal cover $\tilde X$. The above inequality is a refinement of an estimate given by Dranishnikov.

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arXiv2017

On the topological complexity of aspherical spaces

The well-known theorem of Eilenberg and Ganea expresses the Lusternik - Schnirelmann category of an aspherical space as the cohomological dimension of its fundamental group. In this paper we study a similar problem of determining algebraically the topological complexity of the Eilenberg-MacLane spaces. One of our main results states that in the case when the fundamental group is hyperbolic in the sense of Gromov the topological complexity of an aspherical space $K(π, 1)$ either equals or is by one larger than the cohomological dimension of $π\times π$. We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group we establish a vanishing property of this spectral sequence which leads to the main result.

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arXiv2017

Configuration Spaces and Robot Motion Planning Algorithms

The paper surveys topological problems relevant to the motion planning problem of robotics and includes some new results and constructions. First we analyse the notion of topological complexity of configuration spaces which is responsible for discontinuities in algorithms for robot navigation. Then we present explicit motion planning algorithms for coordinated collision free control of many particles moving in Euclidean spaces or on graphs. These algorithms are optimal in the sense that they have minimal number of regions of continuity. Moreover, we describe in full detail the topology of configuration spaces of two particles on a tree and use it to construct some top-dimensional cohomology classes in configuration spaces of n particles on a tree.

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arXiv2014

Fundamental groups of random clique complexes

We study fundamental groups of clique complexes associated to random graphs. We establish thresholds for their cohomological and geometric dimension and torsion. We also show that in certain regime any aspherical subcomplex of a random clique complex satisfies the Whitehead conjecture, i.e. all irs subcomplexes are also aspherical.

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arXiv2011

The topology of spaces of polygons

Let $E_{d}(\ell)$ denote the space of all closed $n$-gons in $\R^{d}$ (where $d\ge 2$) with sides of length $\ell_1,..., \ell_n$, viewed up to translations. The spaces $E_d(\ell)$ are parameterized by their length vectors $\ell=(\ell_1,..., \ell_n)\in \R^n_{>}$ encoding the length parameters. Generically, $E_{d}(\ell)$ is a closed smooth manifold of dimension $(n-1)(d-1)-1$ supporting an obvious action of the orthogonal group ${O}(d)$. However, the quotient space $E_{d}(\ell)/{O}(d)$ (the moduli space of shapes of $n$-gons) has singularities for a generic $\ell$, assuming that $d>3$; this quotient is well understood in the low dimensional cases $d=2$ and $d=3$. Our main result in this paper states that for fixed $d\ge 3$ and $n\ge 3$, the diffeomorphism types of the manifolds $E_{d}(\ell)$ for varying generic vectors $\ell$ are in one-to-one correspondence with some combinatorial objects -- connected components of the complement of a finite collection of hyperplanes. This result is in the spirit of a conjecture of K. Walker who raised a similar problem in the planar case $d=2$.

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arXiv2011

Telescopic Linkages and Topological Approach to Phase Transitions

A topological approach to the theory of equilibrium phase transitions in statistical physics is based on the Topological Hypothesis (TH), which claims that phase transitions are due to changes of the topology of suitable submanifolds in the configuration space. In this paper we examine in detail the anti-ferromagnetic mean-field XY model and study topology of the sub-energy manifolds. The latter can be interpreted mechanically as configuration space of a linkage with one telescopic leg. We use methods of Morse theory to describe explicitly the Betti numbers of this configuration space. We apply these results to the anti-ferromagnetic mean-field XY model and compute the exponential growth rate of the total Betti number. The previous authors instead of the total Betti number studied the Euler characteristic. We show that in the presence of an external magnetic field the model undergoes a single "total Betti number phase transition".

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arXiv2011

Topology of Random Right Angled Artin Groups

In this paper we study topological invariants of a class of random groups. Namely, we study right angled Artin groups associated to random graphs and investigate their Betti numbers, cohomological dimension and topological complexity. The latter is a numerical homotopy invariant reflecting complexity of motion planning algorithms in robotics. We show that the topological complexity of a random right angled Artin group assumes, with probability tending to one, at most three values. We use a result of Cohen and Pruidze which expresses the topological complexity of right angled Artin groups in combinatorial terms. Our proof deals with the existence of bi-cliques in random graphs.

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arXiv2010

Topology of configuration space of two particles on a graph, II

This paper continues the investigation of the configuration space of two distinct points on a graph. We analyze the process of adding an additional edge to the graph and the resulting changes in the topology of the configuration space. We introduce a linking bilinear form on the homology group of the graph with values in the cokernel of the intersection form (introduced in Part I of this work). For a large class of graphs, which we call mature graphs, we give explicit expressions for the homology groups of the configuration space. We show that under a simple condition, adding an edge to a mature graph yields another mature graph.

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arXiv2010

Topology of random 2-complexes

We study the Linial--Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for $p\ll n^{-1}$ a random 2-complex $Y$ collapses simplicially to a graph and, in particular, the fundamental group $π_1(Y)$ is free and $H_2(Y)=0$, a.a.s. We also prove that, if the probability parameter $p$ satisfies $p\gg n^{-1/2+ε}$, where $ε>0$, then an arbitrary finite two-dimensional simplicial complex admits a topological embedding into a random 2-complex, with probability tending to one as $n\to \infty$. We also establish several related results, for example we show that for $p<c/n$ with $c<3$ the fundamental group of a random 2-complex contains a nonabelian free subgroup. Our method is based on exploiting explicit thresholds (established in the paper) for the existence of simplicial embedding and immersions of 2-complexes into a random 2-complex.

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arXiv2010

Random groups arising as graph products

In this paper we study the hyperbolicity properties of a class of random groups arising as graph products associated to random graphs. Recall, that the construction of a graph product is a generalization of the constructions of right-angled Artin and Coxeter groups. We adopt the Erdos - Renyi model of a random graph and find precise threshold functions for the hyperbolicity (or relative hyperbolicity). We aslo study automorphism groups of right-angled Artin groups associated to random graphs. We show that with probability tending to one as $n\to \infty$, random right-angled Artin groups have finite outer automorphism groups, assuming that the probability parameter $p$ is constant and satisfies $0.2929 <p<1$.

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