arXiv subjects

M. Farber

Publications and source records attributed to M. Farber.

15 recordsLinked to original sources

arXiv2015

Large Random Simplicial Complexes, III; The Critical Dimension

In this paper we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in our previous papers of this series. This model includes as special cases the Linial-Meshulam-Wallach model as well as the clique complexes of random graphs. We characterise the concept of critical dimension in terms of various geometric and topological properties of random simplicial complexes such as their Betti numbers, the fundamental group, the size of minimal cycles and the degrees of simplexes. We mention in the text a few interesting open questions.

math.AT↗

arXiv2015

Large Random Simplicial Complexes, II; the fundamental groups

In our recent work we described conditions under which a multi-parameter random simplicial complex is connected and simply connected. We showed that the Betti numbers of multi-parameter random simplicial complexes in one specific dimension dominate significantly the Betti numbers in all other dimensions. In this paper we focus mainly on the properties of fundamental groups of multi-parameter random simplicial complexes, which can be viewed as a new class of random groups. We describe thresholds for nontrivially and hyperbolicity (in the sense of Gromov) for these groups. Besides, we find domains in the multi-parameter space where these groups have 2-torsion. We also prove that these groups have never odd-prime torsion and their geometric and cohomological dimensions are either 0,1, 2 or infinity. Another result presented in this paper states that aspherical 2-dimensional subcomplexes of random complexes satisfy the Whitehead Conjecture, i.e. all their subcomplexes are also aspherical (with probability tending to one).

math.AT↗

arXiv2015

Homological Domination in Large Random Simplicial Complexes

In this paper we state the homological domination principle for random multi-parameter simplicial complexes, claiming that the Betti number in one specific dimension (which is explicitly determined by the probability multi-parameter) significantly dominates the Betti numbers in all other dimensions. We also state and discuss evidence for two interesting conjectures which would imply a stronger version of the homological domination principle, namely that generically homology of a random simplicial complex coincides with that of a wedges of k-dimensional spheres. These two conjectures imply that under an additional assumption (specified in the paper) a random simplicial complex collapses to a k-dimensional complex homotopy equivalent to a wedge of spheres of dimension k.

math.AT↗

arXiv2015

Large random simplicial complexes, I

In this paper we develop further the multi-parameter model of random simplicial complexes. Firstly, we give an intrinsic characterisation of the multi-parameter probability measure. Secondly, we show that in multi-parameter random simplicial complexes the links of simplexes and their intersections are also multi-parameter random simplicial complexes. Thirdly, we find conditions under which a multi-parameter random simplicial complex is connected and simply connected.

math.AT↗

arXiv2015

Random Simplicial Complexes

In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special cases. Topological and geometric properties of a multi-parameter random simplicial complex depend on the whole combination of the probability parameters and the thresholds for topological properties are convex sets rather than numbers (as in all previously known models). We discuss the containment properties, density domains and dimension of the random simplicial complexes.

math.AT↗

arXiv2014

Geometry and Topology of Random 2-complexes

We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that any aspherical subcomplex of a random 2-complex satisfies the Whitehead conjecture. We use inequalities for Cheeger constants and systoles of simplicial surfaces to analyse spheres and projective planes lying in random 2-complexes. Our proofs exploit the strong hyperbolicity property of random 2-complexes.

math.AT↗

arXiv2012

The asphericity of random 2-dimensional complexes

We study random 2-dimensional complexes in the Linial - Meshulam model and prove that for the probability parameter satisfying $$p\ll n^{-46/47}$$ a random 2-complex $Y$ contains several pairwise disjoint tetrahedra such that the 2-complex $Z$ obtained by removing any face from each of these tetrahedra is aspherical. Moreover, we prove that the obtained complex $Z$ satisfies the Whitehead conjecture, i.e. any subcomplex $Z'\subset Z$ is aspherical. This implies that $Y$ is homotopy equivalent to a wedge $Z\vee S^2\vee...\vee S^2$ where $Z$ is a 2-dimensional aspherical simplicial complex. We also show that under the assumptions $$c/n 3$ and $0<ε<1/47$, the complex $Z$ is genuinely 2-dimensional and in particular, it has sizable 2-dimensional homology; it follows that in the indicated range of the probability parameter $p$ the cohomological dimension of the fundamental group $π_1(Y)$ of a random 2-complex equals 2.

math.AT↗

arXiv2004

L^2 torsion without the determinant class condition and extended L^2 cohomology

We associate determinant lines to objects of the extended abelian category built out of a von Neumann category with a trace. Using this we suggest constructions of the combinatorial and the analytic L^2 torsions which, unlike the work of the previous authors, requires no additional assumptions; in particular we do not impose the determinant class condition. The resulting torsions are elements of the determinant line of the extended L^2 cohomology. Under the determinant class assumption the L^2 torsions of this paper specialize to the invariants studied in our previous work. Applying a recent theorem of D. Burghelea, L. Friedlander and T. Kappeler we obtain a Cheeger - Muller type theorem stating the equality between the combinatorial and the analytic L^2 torsions.

math.DG↗

arXiv2003

Smooth Lyapunov 1-forms

We find conditions which guarantee that a given flow on a closed smooth manifold admits a smooth Lyapunov one-form lying in a prescribed de Rham cohomology class. These conditions are formulated in terms of Schwartzman's asymptotic cycles of the flow.

math.DS↗

arXiv2003

Lyapunov 1-forms for flows

In this paper we find conditions which guarantee that a given flow $Φ$ on a compact metric space $X$ admits a Lyapunov one-form $ω$ lying in a prescribed Čech cohomology class $ξ\in \check H^1(X;\R)$. These conditions are formulated in terms of the restriction of $ξ$ to the chain recurrent set of $Φ$. The result of the paper may be viewed as a generalization of a well-known theorem of C. Conley about the existence of Lyapunov functions.

math.DS↗

arXiv2000

Topology of billiard problems, I

Let $T\subset \R^{m+1}$ be a strictly convex domain bounded by a smooth hypersurface $X=\partial T$. In this paper we find lower bounds on the number of billiard trajectories in $T$ which have a prescribed intial point $A\in X$, a prescribed final point $B\in X$ and make a prescribed number $n$ of reflections at the boundary $X$. We apply a topological approach based on calculation of cohomology rings of certain configuration spaces.

math.DG↗

arXiv1999

Morse-Novikov critical point theory, Cohn localization and Dirichlet units

In this paper we construct a Universal chain complex, counting zeros of closed 1-forms on a manifold. The Universal complex is a refinement of the well known Novikov complex; it relates the homotopy type of the manifold, after a suitable noncommutative localization, with the numbers of zeros of different indices which may have closed 1-forms within a given cohomology class. The Main Theorem of the paper generalizes the result of a joint paper with A. Ranicki, which treats the special case of closed 1-forms having integral cohomology classes. The present paper also describes a number of new inequalities, giving topological lower bounds on the number of zeroes of closed 1-forms. In particular, such estimates are provided by the homology of flat line bundles with monodromy described by complex numbers which are not Dirichlet units.

math.DG↗

arXiv1999

On the zero-in-the-spectrum conjecture

We prove that the answer to the "zero-in-the-spectrum" conjecture, in its form, suggested by J. Lott, is negative. Namely, we show that for any n > 5 there exists a closed n-dimensional manifold M, so that zero does not belong to the spectrum of the Laplace-Beltrami operator acting on the L^2 forms of all degrees on the universal covering of M.

math.DG↗

arXiv1996

Equivariant Novikov inequalities

We establish an equivariant generalization of the Novikov inequalities which allow to estimate the topology of the set of critical points of a closed basic invariant form by means of twisted equivariant cohomology of the manifold. We apply these inequalities to study cohomology of the fixed points set of a symplectic torus action. We show that in this case our inequalities are perfect, i.e. they are in fact equalities.

dg-ga↗

arXiv1996

Determinant lines, von Neumann algebras and $L^2$ torsion

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic torsion invariants associated to flat Hilbertian bundles over compact polyhedra and manifolds; we view them as volume forms on the reduced $L^2$ homology and cohomology. These torsion invariants specialize to the the classical Reidemeister-Franz torsion and the Ray-Singer torsion in the finite dimensional case. Under the assumption that the $L^2$ homology vanishes, the determinant line can be canonically identified with $\R$, and our $L^2$ torsion invariants specialize to the $L^2$ torsion invariants previously constructed by A.Carey, V.Mathai and J.Lott. We also show that a recent theorem of Burghelea et al. can be reformulated as stating equality between two volume forms (the combinatorial and the analytic) on the reduced $L^2$ cohomology.

dg-ga↗