. Higher dimensional automata can model concurrent computations. The topological structure of the higher dimensional automata determines certain properties of the concurrent computation. We introduce bicomplexes as an algebraic tool for describing these automata and develop a simple homology theory for higher dimensional automata. We then show how the homology of automata has applications in the study of branching-time equivalences of processes such as bisimulation. 1 Introduction Geometry has been suggested as a tool for modeling concurrency using higher dimensional objects to describe the concurrent execution of processes. This contrasts with earlier models based on the interleaving of computation steps to capture all possible behaviours of a concurrent system. Interleaving models must necessarily commit themselves to a specific choice of atomic action which makes them unable to distinguish between the execution of two truly concurrent actions and two mutually exclusive actions as t...

. We show that the map part of the discrete Conley index carries information which can be used to detect the existence of connections in the repeller-attractor decomposition of an isolated invariant set of a homeomorphism. We use this information to provide a characterization of invariant sets which admit a semi-conjugacy onto the space of sequences on K symbols with dynamics given by a subshift. These ideas are applied to the Henon map to prove the existence of chaotic dynamics on an open set of parameter values. 1; Center for Dynamical Systems and Nonlinear Studies, Georgia Institue of Technology, Atlanta GA 30332. 2 Research funded in part by NSF Grant DMS-9101412 3 Center for Dynamical Systems and Nonlinear Studies, Georgia Institue of Technology, Atlanta GA 30332; on leave from Computer Science Department, Jagiellonian University, Krak'ow, Poland. 1 1. Introduction Examples of complicated or chaotic dynamics are ubiquitous; extending well beyond the mathematical literature...

Abstract: Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational formula and conservation laws for Lagrangian systems on graded manifolds using contact supersymmetries are obtained. 1

Abstract. In this work, the continuously controlled assembly map in algebraic K-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups Γ that satisfy certain geometric conditions. The group Γ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, K0(kΓ) is proved to be isomorphic to the colimit of K0(kH) over the finite subgroups H of Γ, when Γ is a virtually polycyclic group and k is a field of characteristic zero. 1.

Abstract. A new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes, particularly strong for low dimensional sets embedded in high dimensions, is presented. The algorithm runs in linear time. The paper presents the theoretical background of the algorithm, the algorithm itself, experimental results based on an implementation for cubical sets as well as some theoretical complexity estimates. 1.

Abstract. We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an EΓ. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjecture for these groups. Ten years ago, the most popular approach to the Novikov conjecture went via compactifications. If a compact aspherical manifold, say, has a universal cover which suitably equivariantly compactifies, already Farrell and Hsiang [FH] proved that the Novikov conjecture follows. Subsequent work by many authors weakened

Let X be a 1-connected space with free loop space ΛX. We introduce two spectral sequences converging towards H ∗ (ΛX; Z/p) and H ∗ ((ΛX)hT; Z/p). The E2-terms are certain non Abelian derived functors applied to H ∗ (X; Z/p). When H ∗ (X; Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p = 2. AMS subject classification (2000): 55N91, 55P35, 18G50 1

Abstract. Let M be a closed connected manifold, f: M → S 1 be a Morse map, belonging to an indivisible integral class ξ ∈ H 1 (M), v be an f-gradient satisfying the transversality condition. The Novikov construction associates to these data a chain complex C ∗ = C∗(f, v). The first main result of the paper is the construction of a functorial chain homotopy equivalence from C ∗ to the completed simplicial chain complex of the infinite cyclic covering of M, corresponding to ξ. The second main result states that the torsion of this chain homotopy equivalence is equal to the Lefschetz zeta function of the gradient flow for any gradient-like vector field v satisfying the transversality condition and having only hyperbolic closed orbits. 1.

Let Fn be the space of complete flags in k n (where k is R or C). With an arbitrary complete flag f ∈ Fn we associate the standard Schubert cell decomposition Schf of the space Fn whose cells are enumerated by elements from Sn while the dimension over k of such a cell equals the number of inversions in the corresponding

Abstract. Let M be a closed connected manifold, f be a Morse map from M to a circle, v be a gradient-like vector field satisfying the transversality condition. The Novikov construction associates to these data a chain complex C ∗ = C∗(f, v). There is a chain homotopy equivalence between C ∗ and completed simplicial chain complex of the infinite cyclic covering of M. The first main result of the paper is the construction of a functorial chain homotopy equivalence between these two complexes. The second main result states that the torsion of this chain homotopy equivalence equals to the Lefschetz zeta function of the gradient flow for an arbitrary gradient-like vector field v satisfying the transversality condition. 1.