For a local po-space $X$ and a base point $x_0\in X$, we define the universal dicovering space $\Pi:\tilde{X}_{x_0}\to X$. The image of $\Pi$ is the future $\uparrow x_0$ of $x_0$ in $X$ and $\tilde{X}_{x_0}$ is a local po-space such that $|\vec{\pi}_1(\tilde{X},[x_0],x_1)|=1$ for the constant dipath $[x_0]\in\Pi^{-1}(x_0)$ and $x_1\in \tilde{X}_{x_0}$. Moreover, dipaths and dihomotopies of dipaths (with a fixed starting point) in $\uparrow x_0$ lift uniquely to $\tilde{X}_{x_0}$. The fibers $\Pi^{-1}(x)$ are discrete, but the cardinality is not constant. We define dicoverings $P:\hat{X}\to X_{x_0}$ and construct a map $\phi:\tilde{X}_{x_0}\to\hat{X}$ covering the identity map. Dipaths and dihomotopies in $\hat{X}$ lift to $\tilde{X}_{x_0}$, but we give an example where $\phi$ is not continuous.

We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of non-trivial tangling in biomolecules, modeled as alpha complexes.

A local po-space is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known [11] that there are two distinct notions of deformation of higher dimensional automata, "spatial" and "temporal", leaving invariant computer scientific properties like presence or absence of deadlocks. Unfortunately, the formalization of these notions is still unknown in the general case of local po-spaces.

We introduce here a particular kind of local po-space, the "globular CW-complexes", for which we formalize these notions of deformations and which are sufficient to formalize higher dimensional automata. The existence of the category of globular CW-complexes was already conjectured in [11].

After localizing the category of globular CW-complexes by spatial and temporal deformations, we get a category (the category of dihomotopy types) whose objects up to isomorphism represent exactly the higher dimensional automata up to deformation. Thus globular CW-complexes provide a rigorous mathematical foundation to study from an algebraic topology point of view higher dimensional automata and concurrent computations.

e propose a method for calculating cohomology operations on finite simplicial complexes.

Of course, there exist well-known methods for computing (co)homology groups, for example, the "reduction algorithm" consisting in reducing the matrices corresponding to the differential in each dimension to the Smith normal form, from which one can read off the (co)homology groups of the complex [Mun84], or the "incremental algorithm" for computing Betti numbers [DE93]. Nevertheless, little is known about general methods for computing cohomology operations.

For any finite simplicial complex $K$, we give a procedure including the computation of some primary and secondary cohomology operations. This method is based on the transcription of the reduction algorithm mentioned above, in terms of a special type of algebraic homotopy equivalences, called contractions [McL75], of the (co)chain complex of $K$ to a "minimal" (co)chain complex $M(K)$. More concretely, whenever the ground ring is a field or the (co)homology of $K$ is free, then $M(K)$ is isomorphic to the (co)homology of $K$. Combining this contraction with the combinatorial formulae for Steenrod reduced $p$th powers at cochain level developed in [GR99] and [Gon00], these operations at cohomology level can be computed. Finally, a method for calculating Adem secondary cohomology operations $\Phi_q:\; Ker(Sq^2H^q(K))\rightarrow H^{q+3}(K)/Sq^2H^q(K)$ is showed

Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of "geometric" models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, as the "direction" of time matters, we have to allow deformation only as long as we do not reverse the direction of time. This calls for a new homotopy theory: "directed" or di-homotopy. We develop some of the geometric intuition behind this theory and give some hints about the algebraic objects one can associate with it (in particular homology groups). For some historic as well as for some deeper reasons, the theory is at a stage where there is a nice blend between cubical, $\omega$-categorical and topological techniques.

The Curry-Howard correspondence between formulas and types, proofs and programs, proof simplification and program execution, also holds for intuitionistic modal logic S4. It turns out that the S4 modalities translate as a monoidal comonad on the space of proofs, giving rise to a canonical augmented simplicial structure. We study the geometry of these augmented simplicial sets, showing that each type gives rise to an augmented simplicial set which is a disjoint sum of nerves of finite lattices of points, plus isolated $(-1)$-dimensional subcomplexes. As an application, we give semantics of modal proofs (a.k.a., programs) in categories of augmented simplicial sets and of topological spaces, and prove a completeness result in the style of Friedman: if any two proofs have the same denotations in each augmented simplicial model, then they are convertible. This result rests both on the fine geometric structure of the constructed spaces of proofs and on properties of subscone categories--the categorical generalization of the notion of logical relations used in lambda-calculus.

Combinatorial homotopical tools developed in previous works, and consisting essentially of intrinsic homotopy theories for simplicial complexes and directed simplicial complexes, can be applied to explore mathematical models representing images, or directed images, or concurrent processes.

An image, represented by a metric space $X$, can be explored at a variable resolution $\epsilon > 0$, by equipping it with a structure $t_\epsilon X$ of simplicial complex depending on $\epsilon $; this complex can be further analysed by homotophy groups $ \pi \ne (X) = \pi _n(t_\epsilon X)$ and homology groups $ H \ne (X) = H _n(t_\epsilon X). $ Loosely speaking, these objects detect singularities which can be captured by an $n$-dimensional grid, with edges bound by $\epsilon $; this works equally well for continuous or discrete regions of euclidean spaces.

Similarly, a directed image, represented by an "asymmetric metric space", produces a family of directed simplicial complexes $ s_\epsilon X $ and can be explored by the fundamental n-category $ \uparrow \Pi^\epsilon_n (X) $ of the latter. The same directed tools can be applied to combinatorial models of concurrent automata, like Chu-spaces.

The aim of this paper is to provide a short introduction to computational homology based on cubical complexes. The discussed topics include cubical complexes, a reduction algorithm for computing homology of finitely generated chain complexes, and an algorithmic construction of homology of continuous maps via multivalued acyclic representations.

Geometric models have been used by several authors to describe the behaviour of concurrent sytems in computer science. A concurrent computation corresponds to an oriented path (dipath) in a (locally) partially ordered state space, and dihomotopic dipaths correspond to equivalent computations. This paper studies several invariants of the state space in the spirit of those of algebraic topology, but taking partial orders into account as an important part of the structure. We use several categories of fractions of the fundamental category of the state space and define and investigate the related quotient categories of "components". For concurrency applications, the resulting categories can be interpreted as a dramatic reduction of the size of the state space to be considered.