The strong shape category of topological spaces SSh can be defined using the coherent homotopy category CH, whose objects are inverse systems consisting of topological spaces, indexed by cofinite directed sets. In particular, if X, Y are spaces and q : Y → Y is a cofinite HPol-resolution of Y , then there is a bijection between the set SSh(X, Y ) of strong shape morphisms F : X → Y and the set ...

This short note introduces a notion of directed homotopy equivalence (or dihomotopy equivalent) and of “directed” topological complexity (which elaborates on the notion that can be found in e.g. [9]) which have a number of desirable joint properties. In particular, being dihomotopically equivalent implies having bisimilar natural homologies (defined in [5]). Also, under mild conditions, directe...

Let p : E —> B be a locally trivial fiber bundle between closed manifolds where dim E > 5 and B has a handlebody decomposition. A controlled homotopy topological structure (or a controlled structure^ for short) is a map / : M —> E where M is a closed manifold of the same dimension as E and / is a p~ (ε)-equivalence for every ε > 0 (see §2). It is the purpose of this paper to develop an obstruct...

In condensed matter physics and related areas, topological defects play important roles in phase transitions and critical phenomena. Homotopy theory facilitates the classification of such topological defects. After a pedagogic introduction to the mathematical methods involved in topology and homotopy theory, the role of the latter in a number of mainly low-dimensional statistical-mechanical sys...

Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n − 1)-connected (n + 1)-types for n ≥ 0. Introduction The computation of homotopy groups of spheres in low degrees in [Tod62] uses heavily secondary ...