ELEMENTS OF HOMOTOPY THEORY Carlos Prieto∗ ∗ Instituto de Matem´aticas,UNAM 2010 Date of version: January 8, 2010 Preface Topology is qualitative geometry. Ignoring dimensions, several geometric objects give rise to the same topological object. Homotopy theory considers even more geometric objects as equivalent objects. For instance, in homotopy theory, a solid ball of any dimension and a point are considered as equivalent, also a solid torus and a circle are equivalent from the point of view of homotopy theory. This loss of precision is compensated by the effectiveness of the algebraic invariants that are defined in terms of homotopy. The problem of deciding if two spaces are homeomorphic or not is, no doubt, the central problem in topology. We call this the homeomorphism problem. It was not until the creation of algebraic topology that it was possible to give a reasonable answer to such a problem. Now it is not only because of the conceptual simplicity of point-set topology and its adequate symbology, but thanks to the powerful tool provided by algebra and its most convenient functorial relationship to topology that this effectiveness is achieved. For instance, if two spaces have different algebraic invariants, then they cannot be equivalent from the homotopical viewpoint. Therefore, they cannot be homeomorphic either. What we now know as algebraic topology was probably started with the Anal- ysis Situs, Paris, 1895, and its five Compl´ements (Complements), Palermo, 1899; London, 1900; Paris, 1902; Paris, 1902, and Palermo, 1904, of Henri Poincar´e.In the first, he notes that \geometry is the art of reasoning well with badly made figures." And further he says: \Yes, without doubt, but with one condition. The proportions of the figures might be grossly altered, but their elements must not be interchanged and must preserve their relative situation. In other terms, one does not have to worry about quantitative properties, but one must respect the qualitative properties, that is to say precisely those which are the concern of Analysis Situs." Indeed, other works of Poincar´econtain as much interesting topology as the just referred to. This is the case for his memoir on the qualitative theory of differential equations, that includes the famous formula of the Poincar´eindex. This formula describes in topological terms the famous Euler formula, and constitutes one of the first steps in algebraic topology. In these works Poincar´econsiders already maps on manifolds such as, for instance, vector fields, whose indexes determine vii viii Preface the Euler characteristic in his index formula. It is Poincar´etoo who generalizes the question on the classification of manifolds having in mind the classification of the orientable surfaces considered by Moebius in Theorie der elementaren Ver- wandtschaft (Theory of elementary relationship), Leipzig, 1863. This classification problem was also solved by Jordan in Sur la d´eformationdes Surfaces (On the deformation of surfaces), Paris, 1866, who, by classifying surfaces solved an im- portant homeomorphism problem. Jordan studied also the homotopy classes of closed paths, that is, the first notions of the fundamental group, inspired by Riemann, who already had ana- lyzed the behavior of integrals of holomorphic differential forms and therewith the concept of homological equivalence between closed paths. This book has the purpose of presenting the topics that, from my own point of view, are the basic topics of algebraic topology that some way should be learnt by any undergraduate student interested in this area or affine areas in mathematics. The design of the text is as follows. We start with a small Chapter 1, which deals with some basic concepts of general topology, followed by four substantial chapters, each of which is divided into several sections that are distinguished by their double numbering (1.1, 1.2, 2.1, ::: ). Definitions, propositions, theorems, remarks, formulas, exercises, etc., are designated with triple numbering (1.1.1, 1.1.2, ::: ). Exercises are an important part of the text, since many of them are intended to carry the reader further along the lines already developed in order to prove results that are either important by themselves or relevant for future topics. Most of these are numbered, but occasionally they are identified inside the text by italics (exercise). After the chapter on basic concepts, the book starts with the concept of a topological manifold, then one constructs all closed surfaces and stresses the im- portance of algebraic invariants as tools to distinguish topological spaces. Other low dimensional manifolds are analyzed; in particular, one proves that in dimen- sion one the only manifolds are the interval, the circle, the half line, and the line. Then using the Heegaard decomposition, it is shown how 3-dimensional manifolds can be analyzed. We state (with no proof) the important Freedman theorem on the classification of simply connected 4-manifolds and finish presenting other man- ifolds that are important in different branches of mathematics, such as the Stiefel and the Grassmann manifolds. Further on, the elements of homotopy theory are presented. In particular, the mappings of the circle into itself are analyzed introducing the important concept of degree. Homotopy equivalence of spaces is introduced and studied, as a coarser concept than that of homeomorphism. The fundamental group is the first properly algebraic invariant introduced and Preface ix then one gives the proof of the Seifert{van Kampen theorem that allows to compute the fundamental group of a space if one knows the groups of some of its parts. We use this important theorem to compute the fundamental groups of all closed surfaces, as well as of some orientable 3-manifolds, whose Heegaard decomposition is known. Thereafter, covering maps are introduced. This is an essential tool to analyze, from a different point of view, the fundamental group. In the last chapter a short introduction to knot theory is presented, where one sees instances of the usefulness of the several algebraic invariants. On the one hand, the Jones polynomial is presented, and on the other, as an application of the fundamental group, the group of a knot is defined. In this point I want recognize the big impact in this book of all experts that directly or indirectly have influenced my education as a mathematician and as a topologist. At the UNAM Guillermo Torres and Roberto V´azquezwere decisive. Later on, in my doctoral studies in Heidelberg, Germany, I had the privilege of receiving directly the teaching of Albrecht Dold and Dieter Puppe; and indirectly, through several German topology books, of other people, among which I owe a mention to Ralph St¨ocker and Heiner Zieschang's [23], as well as to that of Tammo tom Dieck [7]. Mexico City, Mexico Carlos Prieto Winter 2010 x Table of Contents Contents Preface vii Introduction xiii 1 Basic concepts 1 1.1 Some examples . 1 1.2 Special constructions . 5 1.3 Path connectedness . 13 1.4 Group actions . 18 2 Manifolds 23 2.1 Topological manifolds . 23 2.2 Surfaces . 34 2.3 More low-dimensional manifolds . 48 2.4 Classical groups . 56 3 Homotopy 67 3.1 The homotopy concept . 67 3.2 Homotopy of mappings of the circle into itself . 79 3.3 Homotopy equivalence . 92 3.4 Homotopy extension . 103 3.5 Domain invariance . 109 xi xii Table of Contents 4 The fundamental group 113 4.1 Definition and general properties . 113 4.2 The fundamental group of the circle . 125 4.3 The Seifert{van Kampen theorem . 130 4.4 Applications of the Seifert{van Kampen theorem . 141 5 Covering maps 151 5.1 Definitions and examples . 151 5.2 Lifting properties . 161 5.3 Universal covering maps . 170 5.4 Deck transformations . 179 5.5 Classification of covering maps over paracompact spaces . 184 6 Knots and links 193 6.1 1-manifolds and knots . 193 6.2 Reidemeister moves . 197 6.3 Knots and colors . 200 6.4 Knots, links, and polynomials . 205 6.5 The knot group . 212 References 221 Alphabetical index 222 Symbols 233 Introduction The object of this book is to introduce algebraic topology. But the first ques- tion is, what is topology? This is not an easy question to answer. Trying to define a branch of mathematics in a concise sentence is complicated. However, we may describe topology as that branch of mathematics that studies continuous defor- mations of geometrical objects. The purpose of topology is to classify objects, or at least to give methods in order to distinguish between objects that are not homeomorphic; that is, between objects that cannot be obtained from each other through a continuous deformation. It also provides techniques to study topological structures in objects arising in very different fields of mathematics. The concepts of \deformation," \continuity," and \homeomorphism" will be fundamental through- out the text. However, even not having yet precise definitions, we shall introduce in what follows several examples, that intuitively illustrate those concepts. Figure 0.1 shows three topological spaces; namely, a sphere surface with its poles deleted, another sphere with its polar caps removed (including the polar circles), and a cylinder with its top and bottom removed (including the edges). These three objects can clearly be deformed to each other; hence topology would not distinguish them. Figure 0.1 A sphere with no poles, a sphere with no polar caps and circles, and a cylinder with no edges Figure 0.2 shows the surface of a torus (a \doughnut") and a sphere surface with a handle attached. Each one of these figures is clearly a deformation of the xiii xiv Introduction other. However, it is intuitively clear that the topological object shown in three forms in Figure 0.1 cannot be deformed to the topological object shown in two forms in Figure 0.2.