An Outline of Homology Theory Stephen A. Mitchell June 1997, revised October 2001 Note: These notes contain few examples and even fewer proofs. They are intended only as an outline, to be supplemented by lectures or other sources. 1 Chain Complexes and Exact Sequences A 3-term sequence of abelian groups A !f B !g C is said to be exact if the kernel of g coincides with the image of f. An arbitary sequence (finite or infinite) of abelian groups :::−!An−1−!An−!An+1−!::: is exact if each sequence of three consecutive terms, as shown, is exact. A short exact sequence is an exact sequence of the form 0−!A !f B !g C−!0 Note that f is injective and g is surjective. The next result, known as the \5-lemma" is extremely useful, and elementary to prove. Lemma 1.1 Suppose given a commutative diagram of abelian groups - - - - A1 B1 C1 D1 E1 f g h i j ? ? ? ? ? - - - - A2 B2 C2 D2 E2 1 such that the rows are exact and f; g; i; j are isomorphisms. Then h is an isomorphism. Remark. The proof actually yields a more refined statement, but this is the most commonly used version of the 5-lemma (prove it yourself!). Note also the special case in which the end-terms are zero groups so that the rows are short exact. A chain complex C· is a sequence of abelian groups and group homomor- phisms @n :::−!Cn −! Cn−1−!::: with the property that @n−1@n = 0. In general, the subscript n could range over all integers, but for present purposes we will take n nonnegative; equiv- alently, we assume that all Cn are zero for n negative. The kernel of @n is called the group of n-cycles and denoted ZnC. The image of @n+1 is called the group of n-boundaries (or perhaps n + 1-boundaries; the literature is not consistent) and denoted BnC. Note that Bn ⊂ Zn. The quotient group Zn=Bn is the n-th homology group of C, denoted HnC. Note that a long exact sequence is the same thing as a chain complex all of whose homology groups are zero. Now observe that chain complexes form a category, in an obvious way. A map of chain complexes f : C·−!D· is simply a sequence of group homo- morpisms fn : Cn−!Dn that commutes with the boundary maps: fn - Cn Dn @C @D ? ? - Cn−1 Dn−1 fn−1 Such a map necessarily takes cycles to cycles and boundaries to bound- aries, and therefore induces a map on homology groups HnC−!HnD, de- noted f∗ or Hnf. It is then trivial to check that each homology group Hn defines a functor from the category of chain complexes to the category of abelian groups. Alternatively, we can assemble them all into a single graded 2 1 abelian group H∗C = ⊕n=0HnC, and regard H∗ as a functor to graded abelian groups. A sequence of chain complexes is exact if, at each level n, it is exact as a sequence of abelian groups. f· g· Proposition 1.2 Suppose 0−!A· −! B· −! C·−!0 is a short exact se- quence of chain complexes. Then there is a natural long exact sequence @ :::−!HnA−!HnB−!HnC −! Hn−1A−!::: The homomorphism @ is defined as follows: Let γ 2 HnC be represented by the cycle c 2 Cn. By assumption c = gn(b) for some b 2 Bn (but b need not be a cycle). Since g· is a map of chain complexes, gn−1@Bb = 0 and hence @Bb = fn−1(a) for some a 2 An−1. Since f· is a map of chain complexes, it follows that @Aa = 0|i.e., a is a cycle. Let α denote the homology class of a, and define @γ = α. A somewhat long but easy exercise shows that @ is a well-defined homomorphism, and that the sequence of the proposition is exact. One final remark: if R is any ring, and we replace the term \abelian group" by \R-module" throughout, all definitions, results and proofs in this section go through verbatim. Thus we can talk about chain complexes of R-modules (the homology groups are then R-modules), exact sequences of R-modules, etc. 2 Definition of Singular Homology n n+1 The n-simplex ∆ is the subset of R given by n X ∆ = f(t0; :::; tn): ti ≥ 0; ti = 1g Thus ∆0 is a point, ∆1 is a line segment, ∆2 is a triangle, and so on. In general, ∆n is homeomorphic to the n-disc. Note that the boundary of ∆n is a union of n + 1 copies of ∆n−1. There are canonical coface maps n−1 n i : ∆ −!∆ defined by i(t0; :::; tn−1) = (t0; :::; 0; :::; tn), where the zero is in the i-th slot. Let SnX denote the set of singular n-simplices|that is, the set of all n continuous maps ∆ −!X, and define face maps di : SnX−!Sn−1X by dif = 3 fi. Let CnX denote the free abelian group ZSnX, and define boundary maps @n : CnX−!Cn−1X by n X i @nf = (−1) dif i=0 Proposition 2.1 @n−1@n = 0. The proof is an elementary combinatorial calculation. Hence C:X is a chain complex. The homology groups of this complex are the singular ho- mology groups of X, denoted Hn(X). It is easy to see that the homology groups are covariant functors of X. In fact for each n, X 7! SnX is a functor from spaces to sets, with the induced maps given by composition. Explicitly, if φ : X−!Y is a continuous map, n and f : ∆ −!X is a singular n-simplex of X, Sn(φ)(f) = φf. These functors are compatible with the face maps in an evident way, and we conclude that X 7! C:X is a functor from spaces to chain complexes. Composing with the homology functor from chain complexes to graded abelian groups shows that X 7! H∗(X) is a functor from spaces to graded abelian groups. It follows immediately that homeomorphic spaces have isomorphic homology groups. The proof of the next proposition is an easy exercise. The reason for calling it the Dimension Axiom need not concern us here. Proposition 2.2 Dimension Axiom. Let ∗ denote the space consisting of a single point. Then Hk(∗) is iso- morphic to Z if k = 0 and is zero otherwise. Computing the homology groups of more interesting spaces X requires the machinery of the following section. However we can get some information about low-dimensional homology groups here. Proposition 2.3 H0X is naturally isomorphic to the free abelian group on the set of path-components of X. The proof is an important exercise. Much harder, although still \elemen- tary" is: 4 Proposition 2.4 Let X be a path-connected space with basepoint x0. Then there is a natural homomorphism h : π1(X; x0)−!H1X that induces an iso- morphism from the abelianization of π1(X; x0) to H1X. The homomorphism h is easy to describe. Represent an element of π1(X; x0) by a path λ : I−!X taking both endpoints to the basepoint. Then λ can be regarded as a singular 1-simplex, and as such it is clearly a cycle. Thus h is defined by taking the homotopy class of λ to its homology class. One can show that this yields a well-defined surjective homomorphism, which then automatically factors through the abelianization (π1X)ab. The final and hardest step is to show that (π1X)ab−!H1X is injective. 3 Homotopy Invariance and the Mayer-Vietoris Sequence Homotopy invariance and the Mayer-Vietoris sequence are the two key tech- nical results needed to get the homology machine fired up. They are fairly easy to understand without knowing the proofs. In fact it is not hard to give the rough idea of the proofs, although this is best done in lecture. Theorem 3.1 (Homotopy Invariance) Suppose the maps f; g : X−!Y are homotopic. Then the induced maps on homology coincide: H∗(f) = H∗(g). It follows that if f is a homotopy equivalence, then H∗(f) is an isomor- phism. In other words, homotopy equivalent spaces have isomorphic homol- ogy groups. Now suppose X is the union of two open subsets U,V. Let jU ; jV denote the inclusions of U \V into U,V, respectively. Let iU ; iV denote the inclusions of U,V, respectively, into X. Finally let j : H∗(U \ V )−!H∗U ⊕ H∗V have components H∗(jU );H∗(jV ) and let i : H∗U ⊕H∗V −!H∗X have components H∗(iU ); −H∗(iV ). Note the minus sign in the last definition. Theorem 3.2 (Mayer-Vietoris sequence) Suppose X is the union of two open subsets U,V. Then there are natural homomorphisms @ : HnX−!Hn−1(U \ V ) 5 and a long exact sequence i j @ :::−!Hn(U \ V ) −! HnU ⊕ HnV −! HnX −! Hn−1(U \ V )−!::: Note that if U \V is empty, the theorem implies that H∗X = H∗U ⊕H∗V . This special case is easily proved directly, using the fact that a simplex is a connected space. To prove Theorem 3.2, one might begin by writing down the sequence of chain complexes j0 i0 0−!C∗(U \ V ) −! C∗U ⊕ C∗V −! C∗X−!0 where j0; i0 are the chain level analogues of the homomorphisms j; i defined above. If this sequence was exact, the desired long exact sequence would be obtained immediately from Proposition 1.2.