Simplicial Homology Homology Groups

ϕ 3 ϕ 1 ϕ 1 ker ϕ 2 im im ϕ 1 ϕ 2 ϕ A B C = Fig. 4.11. Groups in Lemma 4.1. ϕ 2 is injective and ϕ 1 is surjective. The operators between the homology groups are induced by the boundary op- erators: We map a homology class to the class of the boundary of one of its members. The Euler characteristic of H ∗ C ∗ , according to the new definition, is simply ∑ i −1 i rank H i = ∑ i −1 i β i . Surprisingly, the homology functor preserves the Euler characteristic of a chain complex. Theorem 4.5 Euler-Poincaré χ C ∗ = χ H ∗ C ∗ . The theorem states that ∑ i −1 i s i = ∑ i −1 i β i for a simplicial complex K, deriving the invariance of the Euler characteristic from the invariance of ho- mology. To prove the theorem, we need a lemma. Lemma 4.1 Let A , B,C be finitely generated Abelian groups related by the sequence of maps ϕ i : ϕ 3 −→ A ϕ 2 −→ B ϕ 1 −→ C ϕ −→ 0, 4.8 where im ϕ i = ker ϕ i − 1 . Then, rank B = rank A + rankC. Proof The sequence is shown in Figure 4.11. First, we establish two facts. a ϕ 1 is surjective: im ϕ 1 = ker ϕ = C. b ϕ 2 is injective: ker ϕ 2 = im ϕ 3 = {e}, so by Corollary 3.1, ϕ 2 is 1-1. By the fundamental homomorphism theorem Theorem 3.10, B ker ϕ 1 ∼ = im ϕ 1 . By fact a, B ker ϕ 1 ∼ = C. Corollary 3.3 gives rank B ker ϕ 1 = rank B − rank ker ϕ 1 , so rankC = rank B − rank ker ϕ 1 . By fact b, A ∼ = im ϕ 2 and rank A = rank im ϕ 2 . But im ϕ 2 = ker ϕ 1 , so rank A = rank ker ϕ 1 . Substituting, we get the desired result. The sequence in the lemma has a name. Definition 4.16 short exact sequence The sequence in Lemma 4.1 is a short exact sequence . We use the lemma to prove the Euler-Poincaré relation. Proof [Euler-Poincaré] Consider the following sequences: −→ Z n i −→ C n ∂ n −→ B n − 1 −→ −→ B n i −→ Z n ϕ −→ H n −→ 0, where 0 is the zero map, i is the inclusion map, and ϕ assigns to a cycle z ∈ Z n its homology class [z] ∈ H n . Both sequences are short exact. Applying Lemma 4.1, we get: rank C n = rank Z n + rank B n − 1 , 4.9 rank Z n = rank B n + rank H n . 4.10 Substituting the second equation into the first, multiplying by −1 n , and sum- ming over n gives the theorem.

4.3 Arbitrary Coefficients

We spent a considerable amount of energy in Sections 3.3.3 and 3.3.4 extend- ing the fundamental theorem of finitely generated Abelian groups to arbitrary R -modules. We now take advantage of our effort to generate additional homol- ogy groups rather quickly. Recall that any finitely generated group is also a Z- module. In this view, we are multiplying elements of a homology group with coefficients from the ring of integers. We may replace this ring with any PID D , such as Z 2 , and the fundamental theorem of finitely generated D-modules Theorem 3.19 would give us a factorization of the homology groups in terms of the module. This fact generates a large number of homology groups, for which we need new notation. Definition 4.17 homology with coefficients The k th homology group with ring of coefficients D is H k K; D = Z k K; D B k K; D. If we choose a field F as set of coefficients, the homology groups become vector spaces with no torsion: H k K; F ∼ = F r , where r is the rank of the vector space. A natural question is whether homology groups generated with differ- ent coefficients are related. The Universal Coefficient Theorem for Homology answers in the affirmative, relating all types of homology to Z homology. Be- fore stating the theorem, we need to look at two new functors that the theorem uses. I will not define these functors formally, as they are large and very inter- esting topics by themselves. Rather, I aim here to state the properties of these functors that allow us to understand the theorem and use it for computation.