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Algebraic Topology Notes Algebraic Topology IV, Michaelmas Term James J. Walton Preface These notes cover the material of the first term of Algebraic Topology IV. The central goal is the construction of the homology groups of a space, and to establish their main properties along with a few applications and computations. You will go deeper into the theory in the second term where you shall also meet cohomology. Along with the notes here, I can suggest Armstrong [Arm] as a gentle introduction to some of the earlier areas we will see (and as a refresher on some point-set topology); see it for a nice account of simplicial homology and a hands-on approach to proving its homotopy invariance. Hatcher's book [Hat] must be recommended: it is nicely written, comprehensive and even freely available online. There are plenty of other good classical texts that you can head to, see Spanier [Spa], Fulton [Ful], Dold [Dol] or Bredon [Bre]. I suggest taking a look around. In places these notes are more descriptive, but less concise than they could be. Because this entails a longer set of notes, I shall also provide a more typical distilled version which may be more to your taste in learning the subject, or perhaps used in conjunction with these notes which can then be read when you need the extra detail. The introduction is there just for initial motivation to the subject|in particular, I won't expect knowledge of the homotopy groups in this term (except for a basic understanding of the fundamental group for the Hurewicz Theorem at the notes). The rest of the material can be read linearly, although sometimes results are stated before we have the tools to prove them, which we then return to later. Sometimes where technical details could get in the way of an otherwise simple idea, I have moved them to the appendix. These are there for your interest and completeness but, as for the introduction, these will not constitute details I expect for you to necessarily have internalised. The exception is the proof of the Snake Lemma, which I have hidden in the appendix to give you space to work out the proof on your own first. A previously undefined term will appear in bold within the text when its definition appears nearby. Concepts which either are to be defined later on, or are being described but without necessarily implying that they are central to the course, will sometimes appear in italics. Some but not all of the homework exercises can be found within the text. Usually the notes' exercises will be easier, they are intended to be thought about whilst reading through the notes. Often they will indicate an idea that will be helpful in thinking about a recently introduced topic, or sometimes even ask for a proof of a result to be used later. All illustrations were produced by the author on Inkscape. Finally: feedback, suggestions or criticisms on the notes is encouraged! 1 Round-up of important notation • N the natural numbers (not including zero), N0 the natural numbers including zero, Z the integers, Q the rational numbers, R the real numbers, C the complex numbers. The cyclic group Z{nZ of order n is denoted Z{n. • 0 sometimes means zero, and sometimes means the trivial group of one element, the context making which clear. • I “ r0; 1s the unit interval. • Sn the n-sphere. • Dn the n-disc. • f : X Ñ Y a morphism f with domain X and codomain Y . • A ãÑ B, an inclusion morphism of A into B. • A B, a surjection from A to B (rarely used. Will usually denote a quotient map). • x ÞÑ y is read `x maps to y' when discussing the definition of a function. • g ˝ f the composition of f followed by g. • f ´1 the inverse of a morphism f. • α Xα the disjoint union of spaces Xα, written X Y for just two spaces. • X ˆ Y the product of spaces X and Y . • X²{„ the quotient space of X under the equivalence² relation „. • X{A the quotient space of X given by collapsing the subspace A Ď X to a point. • X – Y , the spaces X and Y are homeomorphic. • f » g, the maps f and g are homotopic. • X » Y , the spaces X and Y are homotopy equivalent. • G – H, the groups G and H are isomorphic. • α Gα the direct sum of groups Gα, written G ` H for just two groups. • C˚, and variants, indicate a chain complex. À • H˚ homology, occasionally H˚pCq to mean the homology of the chain complex C˚. • f7 chain map, sometimes written pfnq. • f˚ indicates an induced map on homology of some chain map. • H˚pKq the simplicial homology of a simplicial complex K, H˚pXq the singular ‚ ‚ homology of a space X, H˚pX q the cellular homology of a CW complex X , H˚pX; Aq the relative homology of a pair of spaces pX; Aq, H˚pXq the reduced homology of a space X (in short, what H˚p´q is depends on what we put into the brackets). r • æj restriction to index j face, with identification of that face with standard sim- plex. 2 Contents 0 Introduction 5 0.1 What is topology? . 5 0.2 What is algebraic topology? . 6 0.3 Homotopy and homology groups . 7 0.4 The Brouwer Fixed Point Theorem, using homology as a `black box' . 7 1 Some foundations 10 1.1 Basic category theory . 10 1.1.1 Categories and examples . 10 1.1.2 Maps between categories: functors . 13 1.2 Homotopy theory . 16 1.2.1 CW complexes . 20 1.2.2 Euler characteristic . 23 1.2.3 Further topics . 23 2 Homological algebra 24 2.1 Geometric motivation . 24 2.2 Chain complexes . 26 2.2.1 Chain complexes and homology . 26 2.2.2 Exact sequences . 28 2.2.3 Maps between chain complexes . 29 2.2.4 Sub-, kernel, image and quotient chain complexes . 30 2.2.5 Chain homotopies . 31 2.3 The Snake Lemma . 33 2.4 Free Abelian groups, rank, split exact sequences and Euler characteristic 37 2.4.1 The rank of an Abelian group . 37 2.4.2 Split exact sequences . 38 2.4.3 Euler characteristic of chain complexes . 39 3 Homology of spaces 41 3.1 Overview . 41 3.2 Simplicial homology . 42 3.2.1 Abstract simplicial complexes . 42 3.2.2 Geometric simplices . 42 3.2.3 Geometric realisation . 43 3.2.4 The simplicial chain complex . 45 3 3.3 Singular homology . 51 3.3.1 What singular homology is and is not good for . 51 3.3.2 Restricting functions on simplices . 52 3.3.3 Definition of singular homology and basic examples . 53 3.3.4 Functorality: induced maps . 56 3.3.5 Homotopy invariance . 57 3.4 Cellular homology . 60 3.4.1 The cellular boundary map, informally . 62 4 Computational machinery for homology 65 4.1 Relative homology . 66 4.1.1 Definition of relative homology and the LES of a pair . 66 4.1.2 Induced maps . 67 4.2 Reduced homology . 69 4.3 Excision . 71 4.3.1 Warm-up: simplicial excision . 71 4.3.2 Excision for singular homology . 72 4.3.3 Proof of Subdivision Lemma 4.3.1 . 74 4.4 Relative homology and quotients . 81 4.4.1 Some applications of the LES of a quotient . 83 4.5 Equating cellular and singular homology . 87 4.5.1 What is the cellular boundary map, really? . 90 4.5.2 Comparing simplicial and cellular homology . 92 4.6 Homotopy invariance of Euler characteristic . 93 4.7 The Mayer{Vietoris sequence . 93 4.7.1 Cellular version . 93 4.7.2 Singular version . 94 4.7.3 Some applications . 95 4.8 Further topics . 99 4.8.1 Eilenberg{Steenrod axioms for homology . 99 4.8.2 Homology with coefficients . 101 4.8.3 Relation between degree one homology and fundamental group . 103 A Proof of the Snake Lemma 106 A.1 Naturality . 109 B Rank and short exact sequences 110 C Geometric realisation 112 C.1 Geometric realisation . 113 C.2 Geometric geometric realisation . 114 4 Chapter 0 Introduction 0.1 What is topology? Topology is the study of abstract spaces which restricts attention to attributes which remain invariant under deformation. It cares not for angles or lengths, but for more `rugged' features. For example, any loop inscribed on the surface of a ball can always be shrunk to a point within that surface, whereas on a torus there are plenty of non- trivial loops which cannot be pulled tight|the same is true for any torus, whether it looks more like the surface of a doughnut or the surface of a coffee cup. The fact that fl – Figure 0.1.1: The 2-sphere, the 2-torus and a coffee mug, whose surface is homeomorphic to the 2-torus. spaces are allowed to be freely deformed may at first seem like a weakness. If one is building a bridge, one cares about angles and lengths! But for some problems, it is natural to strip away some of the geometry to get to the heart of the matter. In finding a path crossing each of the bridges of K¨onigsburgonly once, the lengths of the bridges are irrelevant, all that matters is how many bridges there are and how they connect to each other. If you are investigating how a string is knotted, it doesn't matter what the length of the string is or exactly how you choose to embed it in space.
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