MAS435 Algebraic Topology 2014{15 Part B: Semester 2 Prof Sarah Whitehouse (based on notes taken by Alex Corner from Dr E Cheng's lectures) Contents Introduction 2 1 Chain complexes and homology 3 1.1 Chain complexes . .3 1.2 Homology . .4 1.3 Products of abelian groups . .5 1.4 Free abelian groups . .5 1.5 Maps between chain complexes . .6 2 Homology of cell complexes and ∆-complexes 7 2.1 Cell complexes and homology . .7 2.2 Low-dimensional examples . .8 2.3 Simplices . 10 3 Singular homology and homotopy invariance 13 3.1 Reduced homology . 13 3.2 Homotopy invariance . 14 4 Abelian groups and abelianisation 18 4.1 Abelianisation . 18 4.2 Finitely-generated abelian groups . 19 5 What happened to van Kampen's Theorem? 22 6 Quotients and relative homology 27 6.1 Excision and quotients . 27 6.2 The Mayer-Vietoris Sequence . 31 7 Axioms for homology 33 8 Further remarks 34 8.1 Wedge sums . 34 8.2 Suspension . 35 CONTENTS 2 8.3 Moore spaces . 36 9 Exercises 37 Introduction We have seen that algebraic topology is about studying topological spaces using algebra. In the first part of the course the algebra we used was group theory. We saw how to study a topological space X via its \fundamental group" π1X which helped us study some features of spaces but not others. The main limita- tion was that the fundamental group is constructed from loops in a space, and cannot detect higher-dimensional features. The higher-dimensional versions of the fundamental group are constructed from higher-dimensional loops. That is, where a loop in X is a map S1 −! X a higher-dimensional loop is a map Sn −! X for higher values of n; homotopy classes of these loops form a group called the nth homotopy group πnX. The trouble with this approach is that higher homotopy groups are very difficult to compute. That is why we turn to homology instead. It is easier to compute than higher homotopy groups, but as a trade-off, it is less sensitive. Homology is the main topic for this semester. We will see a couple of ways to define it and we will study its properties. For calculations, we will mainly stick to fairly low-dimensional examples, but the methods we cover apply in higher-dimensional situations. The idea of homology is to produce, from a space X, a sequence of abelian groups Hn(X), where n ≥ 0. The group Hn(X) is called the nth homology group of X. In fact, we proceed in two steps: space chain complex homology groups A \chain complex" is a well-behaved sequence of abelian groups and homomor- phisms, as we'll see. This means that homology can be used to study a wide 1 Chain complexes and homology 3 range of things, not just spaces, as long as some sort of chain complex can be obtained from those things. So the first step above immediately gets us into algebra, and the second step is all within the world of algebra. The study of chain complexes and their associated homology groups is called homological algebra and can be thought of as a particularly structured part of the world of abelian groups. For the first step, it helps if we know how our space is built up dimension by dimension from \cells", that is, some form of disc/ball. Homology will then 1. find the places where we could have attached discs, and 2. compare it with the places where we actually did attach discs. Formally, this is done by taking a quotient group. This quotient group measures how \holey" the space is, because it measures the \holes" that were not filled in by discs. The slogan of homology is cycles mod boundaries. A \cycle" is the algebraic version of a \hole". A \boundary" is a hole that we filled in with a disc - it has now become the boundary of a disc. 1 Chain complexes and homology We need to start by looking at the algebraic objects we'll be using: chain com- plexes of abelian groups. All the groups in this part of the course are abelian. 1.1 Chain complexes Definition 1.1. A chain complex is a sequence of abelian groups and group homomorphisms δn+2 δn+1 δn δ2 δ1 δ0 ··· Cn+1 Cn Cn−1 ··· C2 C1 C0 0 such that for all n δnδn+1 = 0: Equivalently Im δn+1 ⊆ ker δn, (and this is an inclusion of subgroups). Exercise 1.2. Show that these conditions really are equivalent. 1.2 Homology 4 Terminology The maps δn are called boundary maps or differentials and a chain complex such as the one above may be labelled as C• or C∗ or sometimes just C. The elements of Im δn+1 are called boundaries and those of ker δn are called cycles. Definition 1.3. If Imδn+1 = ker δn for all n then the chain complex is called an exact sequence. Another slogan is that homology measures the failure of a chain complex to be exact, as we will now see from the definition. 1.2 Homology Definition 1.4. The nth homology group of a chain complex C is defined to be the quotient group Hn(C) = ker δn= Im δn+1: Exercise 1.5. What are two different ways of thinking of the elements of this group? Do we have to worry about whether or not Im δn+1 is a normal subgroup of ker δn ? Thinking of elements of Hn(C) as equivalence classes of cycles, we call them ho- mology classes. Two equivalent cycles are then called homologous; the condition for being homologous is, for f; g 2 ker δn, f ∼ g , f − g 2 Im δn+1: Exercise 1.6. Find the homology groups of the following chain complexes. (In each diagram the rightmost arrow is δ0.) 0 ×2 0 0 0 Z Z Z 0 0 0 0 0 0 Z Z Z 0 0 1 1 0 0 Z Z Z 0 0 δn = 0 0 0 0 Z 0 0 ··· 0 0 1.3 Products of abelian groups 5 1.3 Products of abelian groups We will be taking many products of abelian groups so it is important to be clear about how this works. The fact that the groups are abelian is crucial here - non-abelian groups work quite differently. (Abelian groups are easier.) Recall that given two abelian groups A and B we can form their direct product A × B, elements of which are pairs (a; b) where a 2 A and b 2 B. This is again an abelian group, inheriting its group operation from A and B: (a; b) + (a0; b0) = (a + a0; b + b0): As A × B is abelian here, the group operation will be written additively. Exercise 1.7. Show that A × B really is abelian. Similarly, recall the direct sum of A and B, the abelian group A ⊕ B. This has elements a ⊕ b where again a 2 A and b 2 B. The group operation is given by (a ⊕ b) + (a0 ⊕ b0) = (a + a0) ⊕ (b + b0), again using the group operations of A and B. There is then an obvious isomorphism of abelian groups A × B =∼ A ⊕ B. Exercise 1.8. Construct this isomorphism. You should construct the map, show it's a group homomorphism, and show it's an isomorphism. Remark 1.9. Note that the direct sum A⊕B is the coproduct of A and B in the category Ab of abelian groups and group homomorphisms. This is not the case for arbitrary groups in the category Gp of groups and group homomorphisms. In Gp the coproduct of groups G and H is the free product G ∗ H. However in both Ab and in Gp the product of two groups is the same, i.e. it is the direct product. In Ab, finite products and coproducts are essentially the same (but infinite ones are not). 1.4 Free abelian groups The free abelian group on one generator is isomorphic to Z. For two generators it is isomorphic to Z ⊕ Z. Carrying on in this way, for k generators, it is ⊕k isomorphic to Z . If we write the generators/basis elements as e1; : : : ; ek then ⊕k k a general element of Z is Σi=1λiei, where λi 2 Z. All of our chain complexes will involve free abelian groups generated by cells of the space. Example 1.10. Consider the chain complex 1.5 Maps between chain complexes 6 δ2 δ1 δ0 0 Z ⊕ Z Z 0 α a β a where δ2 and δ0 are the zero maps and δ1 maps the generators of Z⊕Z as shown. Notice that ker δ0 = Z and Imδ1 = Z, both generated by a, so H0 = Z=Z = 0. Also notice that the kernel of δ1 is generated by one element, α − β, and that Im δ2 = 0, thus H1 = Z=0 = Z. The higher homology groups, for n ≥ 2, are all the trivial group 0. 1.5 Maps between chain complexes Just as there are maps between abelian groups, we can define maps between chain complexes. Definition 1.11. A chain map f : A ! B between chain complexes A and B is a collection of group homomorphisms fn : An ! Bn, for all n, such that the following diagram commutes for all n. δn An An−1 fn fn−1 Bn Bn−1 n Proposition 1.12. A chain map f : A ! B induces a group homomorphism Hn(f): Hn(A) ! Hn(B), for all n.