A combinatorial model for the free loop fibration

We introduce the abstract notion of a closed necklical set in order to describe a functorial combinatorial model of the free loop fibration $\Omega Y\rightarrow \Lambda Y\rightarrow Y$ over the geometric realization $Y=|X|$ of a path connected simplicial set $X.$ In particular, to any path connected simplicial set $X$ we associate a closed necklical set $\widehat{\mathbf{\Lambda}}X$ such that its geometric realization $|\widehat{\mathbf{\Lambda}}X|$, a space built out of gluing "freehedrical" and "cubical" cells, is homotopy equivalent to the free loop space $\Lambda Y$ and the differential graded module of chains $C_*(\widehat{\mathbf{\Lambda}}X)$ generalizes the coHochschild chain complex of the chain coalgebra $C_\ast(X).$Comment: Made minor revisions. To appear in Bulletin of the London Mathematical Societ

Algebraic Topology

The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook \emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael Grinfeld from the University of Strathclyd

A geometry of information, I: Nerves, posets and differential forms

The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial Representation: Continuous vs. Discrete'. Spatial representation has two contrasting but interacting aspects (i) representation of spaces' and (ii) representation by spaces. In this paper, we will examine two aspects that are common to both interpretations of the theme, namely nerve constructions and refinement. Representations change, data changes, spaces change. We will examine the possibility of a `differential geometry' of spatial representations of both types, and in the sequel give an algebra of differential forms that has the potential to handle the dynamical aspect of such a geometry. We will discuss briefly a conjectured class of spaces, generalising the Cantor set which would seem ideal as a test-bed for the set of tools we are developing.Comment: 28 pages. A version of this paper appears also on the Dagstuhl seminar portal http://drops.dagstuhl.de/portals/04351

Explicit Simplicial Discretization of Distributed-Parameter Port-Hamiltonian Systems

Simplicial Dirac structures as finite analogues of the canonical Stokes-Dirac structure, capturing the topological laws of the system, are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators. These finite-dimensional Dirac structures offer a framework for the formulation of standard input-output finite-dimensional port-Hamiltonian systems that emulate the behavior of distributed-parameter port-Hamiltonian systems. This paper elaborates on the matrix representations of simplicial Dirac structures and the resulting port-Hamiltonian systems on simplicial manifolds. Employing these representations, we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port-Hamiltonian systems on simplicial manifolds

Chain Homotopies for Object Topological Representations

This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AM-model. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AM-model and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here. A concept of generators which are "nicely" representative cycles is also presented. Moreover, we extend the definition of AM-models to 3D binary digital images and we design algorithms to update the AM-model information after voxel set operations (union, intersection, difference and inverse)

Complex Line Bundles over Simplicial Complexes and their Applications

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of Andr\'e Weil on the classification of hermitian line bundles. Moreover, we associate to each discrete hermitian line bundle with curvature a unique piecewise-smooth hermitian line bundle of piecewise constant curvature. This is then used to define a discrete Dirichlet energy which generalizes the well-known cotangent Laplace operator to discrete hermitian line bundles over Euclidean simplicial manifolds of arbitrary dimension

Spatial Topology and its Structural Analysis based on the Concept of Simplicial Complex

This paper introduces a model that identifies spatial relationships for a structural analysis based on the concept of simplicial complex. The spatial relationships are identified through overlapping two map layers, namely a primary layer and a contextual layer. The identified spatial relationships are represented as a simplical complex, in which simplices and vertices respectively represent two layers of objects. The model relies on the simplical complex for structural representation and analysis. To quantify structural properties of individual primary objects (or equivalently simplices), and the simplicial complex as a whole, we define a set of centrality measures by considering multidimensional chains of connectivity, i.e. the number of contextual objects shared by a pair of primary objects. With the model, the interaction and relationships with a geographic system are modeled from both local and global perspectives. The structural properties and modeling capabilities are illustrated with a simple example and a case study applied to the structural analysis of an urban system.Comment: 14 pages, 7 figures, 2 tables, submitted for publicatio