Rational homotopy type of subspace arrangements

Remerciements La première personne que je souhaite remercier est évidemment mon cher promoteur, Yves Félix. C'est grâce à sa patience et à sa vision que j'ai appris et apprécié le monde de la topologie algébrique. Une autre personne qui a joué un rôle important dans mon éducation mathématique est certainement Pascal Lambrechts, qui mérite toute ma gratitude pour m'avoir enseigné tellement de choses. Je souhaite également remercier les membres de mon jury pour leur lecture attentive de cette thèse : trop vite, et sa " vindicte " quotidienne me manquera certainement! à ma famille, ainsi qu'à tous ceux que j'ai oublié de citer, pour les bons moments passés ensembles, pendant les études et/ou après. Et last but not least, merci à ma petite kartoffel, Anneliese. iii iv Contents Introduction vii Chapter 1 Homotopy and rational homotopy theory 1 1. An arrangement of hyperplanes is a finite set of hyperplanes (affine subspace of codimension one) in some finite dimensional vector space V (in practice, it will often be R m or C m). The study of this area of mathematics started with the cheese cutting problem : What is the largest number of pieces that we can obtain by cutting a cheese with n cuts? (see figure 1) Mathematically, this question can be reformulated in the following way : among the finite sets of n planes in R 3 , what is the largest number of connected components? This simple problem is one of the earliest example of a question in the subject of arrangements of hyperplanes. Figure 1: Cheese cut with three planes The cheese cutting problem is a typical example : with some combinatorial data (number of hyperplanes, dimension of V), we want to know a topological property of the complement space : its number of connected components. A complete solution was given by vii viii Introduction L. Schläfli in 1901 (see [29]). Also, this problem is discussed in chapter 4. Since then, and for various reasons, many arrangements of hy-perplanes appeared in different areas of science and were studied by mathematicians. As illustrated by the cheese cutting problem, the earlier apparitions of arrangements involved partition problems in Euclidean and projective spaces. But it was discovered that the notion of arrangement of hyperplanes connects many aspects of mathematics which seemed to be quite unrelated, in particular, topology, geometry and combinatorics. In 1971, B. Grünbaum published a survey of the …