Christian Bär Algebraic Topology Lecture Notes, Winter Term 2016/17 OTSDAM P EOMETRIE IN G Version of February 23, 2017 © Christian Bär 2017. All rights reserved. Picture on title page taken from www.sxc.hu. Contents Preface 1 1 Set Theoretic Topology 3 1.1 Typical problems in topology ........................... 3 1.2 Some basic definitions .............................. 7 1.3 Compactness ................................... 9 1.4 Hausdorff spaces ................................. 10 1.5 Quotient spaces .................................. 11 1.6 Product spaces .................................. 12 1.7 Exercises ..................................... 13 2 Homotopy Theory 17 2.1 Homotopic maps ................................. 17 2.2 The fundamental group .............................. 21 2.3 The fundamental group of the circle ....................... 28 2.4 The Seifert-van Kampen theorem ......................... 40 2.5 The fundamental group of surfaces ........................ 52 2.6 Higher homotopy groups ............................. 58 2.7 Exercises ..................................... 75 3 Homology Theory 79 3.1 Singular homology ................................ 79 3.2 Relative homology ................................ 84 3.3 The Eilenberg-Steenrod axioms and applications ................ 87 3.4 The degree of a continuous map ......................... 96 3.5 Homological algebra ............................... 105 3.6 Proof of the homotopy axiom ........................... 112 3.7 Proof of the excision axiom ............................ 116 3.8 The Mayer-Vietoris sequence ........................... 123 3.9 Generalized Jordan curve theorem ........................ 126 3.10 CW-complexes .................................. 130 3.11 Homology of CW-complexes ........................... 135 3.12 Betti numbers and the Euler number ....................... 140 3.13 Incidence numbers ................................ 143 3.14 Homotopy versus homology ........................... 146 3.15 Exercises ..................................... 152 Sources list for pictures 157 i Contents Bibliography 159 Index 161 ii Preface These are the lecture notes of an introductory course on algebraic topology which I taught at Potsdam University during the winter term 2016/17. The aim was to introduce the basic tools from homotopy and homology theory. Choices concerning the material had to be made. Since time was too short for a reasonable discussion of cohomology theory after homology had been treated, I decided to skip cohomology altogether and instead included more material from homotopy theory than is often done. In particular, there is a detailed discussion of higher homotopy groups and the long exact sequence for Serre fibrations. The necessary prerequisites of the students were rather modest. The course contains a quick introduction to set theoretic topology but a certain acquaintance with these concepts was certainly helpful. Familiarity with basic algebraic notions like rings, modules, linear maps etc. was assumed. These notes are based on the lecture notes of a course I taught in 2010. I am grateful to Volker Branding who wrote a first draft of those lecture notes and produced most of the pstricks-figures and to Ramona Ziese who improved those notes considerably. Potsdam, February 2017 Christian Bär 1 1 Set Theoretic Topology 1.1 Typical problems in topology Topology is rough geometry. For example, a sphere is topologically the same as a cube, even though the sphere is smooth and curved while the cube is piecewise flat and has corners. In more precise mathematical terms this means that they are homeomorphic. On the other hand, the sphere is different from a torus even topologically. ≈ 0 Here are four versions of a typical question that one asks in mathematics. Fix the integers 1 n < m. ≤ 1.) Does there exist a linear isomorphism ϕ : Rm Rn ? → No, since linear algebra tells us that isomorphic vector spaces have equal dimension. 2.) Does there exist a diffeomorphism ϕ : Rm Rn ? → No, otherwise the differential dϕ(0) : Rm Rn would be a linear isomorphism. → 3.) Does there exist a homeomorphism ϕ : Rm Rn ? → We cannot answer that question yet, but we will develop the necessary tools to find the answer. 4.) Does there exist a bijective map ϕ : Rm Rn ? → Yes. We will now explicitly construct an example for such a map in the case n = 1 and m = 2. Example 1.1. Since the exponential function maps R bijectively onto R+ = (0, ) it suffices to ∞ construct a bijective map ϕ : R+ R+ R+. → × 3 1 Set Theoretic Topology Given x > 0 write x as infinite decimal fraction. x = ... a3a2a1, b1b2b3 ... Here aj 0, 1,..., 9 and almost all aj are equal to 0. The bj are blocks of the form 0 ... 0cj ∈ { } with cj 1,..., 9 . This representation of x is unique. Now define ϕ(x) = (ϕ (x),ϕ (x)) ∈ { } 1 2 where ϕ1(x) = ... a5a3a1, b1b3b5 ... ϕ2(x) = ... a6a4a2, b2b4b6 ... One easily checks that ϕ maps R+ bijectively onto R+ R+. × Let us evaluate the construction for an example. Let x = 1987, 30500735 .... Then we can read off the aj and the bj as a = 7, a = 8, a = 9, a = 1, aj = 0 for j 5 1 2 3 4 ≥ b1 = 3, b2 = 05, b3 = 007, b4 = 3, b5 = 5,... Hence ϕ1(x) = 97, 30075 ... and ϕ2(x) = 18, 053 ... Remark 1.2. In the continuous world counter-intuitive things can happen which are not possible in the differentiable world. For example, there exist continuous maps ϕ : [0, 1] [0, 1] [0, 1] → × which are surjective. Such a map is called a plane-filling curve. Example 1.3. The first example goes back to Peano [6]. The following example was shortly after given by Hilbert [3] and is known as the Hilbert curve. It is defined by ϕ(x) := lim ϕn (x) n →∞ where the ϕn are defined recursively as indicated by Hilbert in the pictures: 4 1.1 Typical problems in topology ϕ1 ϕ2 ϕ3 ϕ4 ϕ5 ϕ6 Remark 1.4. This cannot happen for smooth curves due to a Theorem by Sard which states that for smooth ϕ : [0, 1] R2 the image ϕ([0, 1]) R2 is a zero set. → ⊂ Many typical problems in topology are of the following form: 1.) Given two spaces, are they homeomorphic? To show that they are, construct a homeomor- phism. To show that they are not, find topological invariants, which are different for given spaces. 2.) Classify all spaces in a certain class up to homeomorphisms. 3.) Fixed point theorems. Example 1.5 (Classification theorem for surfaces). The classification theorem for surfaces states that each orientable compact connected surface is homeomorphic to exactly one in the following infinite list: 5 1 Set Theoretic Topology S2 (sphere), genus 0 T2 (torus), genus 1 F2 (Pretzel surface), genus 2 F3 (true pretzel), genus 3 b b b The genus is the number of “holes” in the surface. We will give a more precise definition later. So the classification theorem says that to each g N there exists an orientable compact connected ∈ 0 surface Fg with genus g and each orientable compact connected surface is homeomorphic to exactly one Fg. Example 1.6 (Brouwer fixed point theorem). Any continuous map f : Dn Dn with → Dn = x Rn x 1 has a fixed point, i.e., there exists an x Dn such that { ∈ |k k ≤ } ∈ f (x) = x. We give a proof for n = 1. Consider the continuous function g : [ 1, 1] R defined by − → g(x) := f (x) x. Now f 1 implies − | | ≤ g( 1) 1 ( 1) = 0, g(1) 1 1 = 0. − ≥− − − ≤ − By the intermediate value theorem we can find an x with g(x) = 0 which is equivalent to f (x) = x. 6 1.2 Some basic definitions 1.2 Some basic definitions First of all let us recall the following Definition 1.7. A subset U Rn is called open iff ⊂ x U r > 0 : B(x, r) U ∀ ∈ ∃ ⊂ n r where B(x, r) = y R d(x, y) = x y < r . b x { ∈ | k − k } The set of open subsets of Rn is called the standard topology of Rn. We write n n Rn := U R open (R ). T { ⊂ } ⊂ P One easily checks Proposition 1.8. Rn has the following properties T n (i) , R Rn ∅ ∈ T (ii) Uj Rn, j J j J Uj Rn ∈T ∈ ⇒ ∈ ∈ T (iii) U , U Rn U S U Rn 1 2 ∈T ⇒ 1 ∩ 2 ∈T The importance of the concept of open subsets comes from the fact that one can characterize continuous maps entirely in terms of open subsets. Namely, a map f : Rn Rm is continuous 1 1 → iff for each U Rm the preimage f (U) is open, f (U) Rn . This motivates taking open ∈ T − − ∈ T subsets axiomatically as the starting point in topology. Definition 1.9. A topological space is a pair (X, X ) with X (X) such that T T ⊂ P (i) , X X ∅ ∈ T (ii) Uj X, j J j J Uj X ∈T ∈ ⇒ ∈ ∈ T (iii) U , U X U S U X 1 2 ∈T ⇒ 1 ∩ 2 ∈ T 7 1 Set Theoretic Topology Definition 1.10. Let (X, X ) be a topological space. Elements of X are called open subsets T T of X. Subsets of the form X U with U X are called closed. \ ∈T Examples 1.11. 1.) X arbitrary set, X = (X) (discrete topology). All subsets of X are open T P and they are also all closed. 2.) X arbitrary set, X = , X (coarse topology). Only X and are open subsets of X. They T {∅ } ∅ are also the only closed subsets. 3.) Let (X, X ) be a topological space, let Y X be an arbitrary subset. The induced topology T ⊂ or subspace topology of Y is defined by Y := U Y U X . T { ∩ | ∈ T } 4.) Let (X, d) be a metric space. Then we can imitate the construction of the standard topology on Rn and define the induced metric as the set of all U X such that for each x U there ⊂ ∈ exists an r > 0 so that B(x, r) U. Here B(x, r) = y X d(x, y) < r is the metric ball ⊂ { ∈ | } of radius r centered at x.