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Introduction to Combinatorial Homotopy Theory

1 Introduction. Homotopy theory is a subdomain of topology where, instead of considering the category of topological spaces and continuous maps, you prefer to consider as morphisms only the continuous maps up to homotopy, a notion precisely defined in these notes in Section 4. Roughly speaking, you decide not to distinguish two maps which can be continuously deformed into each other; such a weakening of the notion of map is quickly identified as necessary when you intend to apply to Topology the methods of Algebraic Topology. Otherwise the main classification problems of topology are, except in low dimensions, out of scope. If you want to " algebraize " the topological world, you will meet another difficulty. The traditional topological spaces, defined for example through collections of open subsets, cannot be directly processed by a computer; a computer can handle only discrete objects and in a sense topology is the opposite subject. A combina-torial intermediary notion between Topology and Algebra is required. Poincaré started Algebraic Topology about a century ago by using the polyhedra as intermediary objects, but since the fifties, the simplicial notions have been recognized as more appropriate. In this framework of combinatorial topology, the sensible topo-logical spaces can be combinatorially defined, and also installed and processed on a computer. This is valid even for very complicated or abstract spaces such as classifying spaces, functional spaces; the various important functors of algebraic topology can also be implemented as functional objects. In a sense there is a conflict between both previous observations. The homotopy relation is concerned by continuous deformations of maps, while combinatorial models for topological spaces and maps do not seem to allow enough maps to model homotopies. But we will see this apparent obstacle is easily overcome, and the so-called Combinatorial Homotopy Theory is now one of the standard ground theories for Algebraic Topology. In particular, if you claim you are mainly interested by constructive results in Algebraic Topology, it is quickly obvious combinatorial topology is required. Constructive Algebraic Topology is a difficult but fascinating subject, and three main solutions are now available: 1 1. Rolf Schön's solution [21], quite elegant, unfortunately never (?) considered since his remarkable memoir, in particular from a concrete programming point of view. 2. The solution studied for years by this author and several collaborators, see the lecture notes of the previous Map Summer School at Genova [20]. The key point is …