Abstract:
It is shown that the Novikov inequalities for critical points of closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a corollary we obtain a vanishing theorem for $L^2$ cohomology, generalizing a theorem of W. Lueck. We also prove that von Neumann Betti numbers coincide with the Novikov numbers for free abelian coverings.

Abstract:
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.

Abstract:
Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters -- the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R^3. We also prove results about higher moments of Betti numbers.

Abstract:
It is shown that for any piecewise-linear closed orientable manifold of odd dimension there exists an invariantly defined metric on the determinant line of cohomology with coefficients in an arbitrary flat bundle E over the manifold (E is not required to be unimodular). The construction of this metric (called Poincare - Reidemeister metric) is purely combinatorial; it combines the standard Reidemeister type construction with Poincare duality. The main result of the paper states that the Poincare-Reidemeister metric computes combinatorially the Ray-Singer metric. It is shown also that the Ray-Singer metrics on some relative determinant lines can be computed combinatorially (including the even-dimensional case) in terms of metrics determined by correspondences.

Abstract:
It is shown that the topological phenomenon "zero in the continuous spectrum", discovered by S.P.Novikov and M.A.Shubin, can be explained in terms of a homology theory on the category of finite polyhedra with values in certain abelian category. This approach implies homotopy invariance of the Novikov-Shubin invariants. Its main advantage is that it allows to use the standard homological techniques, such as spectral sequences, derived functors, universal coefficients etc., while studying the Novikov-Shubin invariants. It also leads to some new quantitative invariants, measuring the Novikov-Shubin phenomenon in a different way, which are used in order to strengthen the Morse type inequalities of Novikov and Shubin.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
In this paper we study a new topological invariant $\Cat(X,\xi)$, where $X$ is a finite polyhedron and $\xi\in H^1(X;\R)$ is a real cohomology class. $\Cat(X,\xi)$ is defined using open covers of $X$ with certain geometric properties; it is a generalization of the classical Lusternik -- Schnirelman category. We show that $\Cat(X,\xi)$ depends only on the homotopy type of $(X,\xi)$. We prove that $\Cat(X,\xi)$ 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 $\Cat(X,\xi)$, which uses cup-products of cohomology classes of flat line bundles with monodromy described by complex numbers, which are not Dirichlet units.

Abstract:
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 $cat(X,\xi)$ of the notion of Lusternik - Schnirelman category, depending on a topological space $X$ and a cohomology class $\xi\in H^1(X;\R)$. We prove that any closed 1-form has at least $cat(X,\xi)$ zeros assuming that it admits a gradient-like vector field with no homoclinic cycles. We show that the number $cat(X,\xi)$ 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.