We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk and A. Weil, which state that the nerve of a regular cover is homotopy equiva...

We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of this is introduction group structures on spaces based loops a smooth manifold, relying certain homotopy equivalence relations -- such as so-called thin and resulting interpretation fields homomorphisms Lie $G$ satisfying suitable smoothness condition, encoding orbit connections principal $G$-b...

Under certain homological hypotheses on a compact 4–manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. The main examples are the class of finite connected sums of 4–manifolds with certain product geometries. Most of these compact manifolds have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which a...

Motivated by constructions in topological data analysis and algebraic combinatorics, we study homotopy theory on the category of Čech closure spaces Cl, whose objects are sets endowed with a operator morphisms continuous maps between them. We introduce new classes structures metric spaces, graphs, simplicial complexes, show how each these cases gives rise to an interesting theory. In particular...