Chapter 1 - Homotopy Types

Abstract

This chapter provides an overview of homotopy types. The theory of homotopy types is one of the most basic parts of topology and geometry. At the center of this theory stands the concept of algebraic invariants. Homotopy types are equivalence classes, homotopy types, and homeomorphism types.. To this end, one uses the notion of deformation or homotopy. The principal idea is to consider “nearby” objects (that is, objects, which are “deformed” or “perturbed” continuously a little bit) as being similar. This idea of perturbation is a common one in mathematics and science. The properties that remain valid under small perturbations are considered the stable and essential features of an object. Homotopy types of polyhedra are archetypes underlying most geometric structures. There are many different topological and combinatorial devices that can be used to construct the homotopy types of connected polyhedra, for example, simplicial complexes, simplicial sets, CW-complexes, topological spaces, simplicial groups, small categories, and partially ordered sets.