## § Snake lemma

### § Why homomorphisms for chain maps?

First of all, to define a mapping between simplicial complexes $\{ G_i \}$ and $\{ H_i \}$, one might naively assume that we can ask for functions $\{ f_i: G_i \rightarrow H_i \}$:
       ∂    ∂
G3 → G2 → G1 → 0
|    |    |
f    g    h
↓    ↓    ↓
0 → H3 → H2 → H1
∂    ∂

Unfortunately, to be able to use the machinery of Homology, we need the $\{ f_i \}$ to be abelian group homomorphisms. However, this is no great loss. Intuitively, when we want to map complexes, we first say where the generators of the abelian group ( $\mathbb Z$-module) maps to; Everything else is determined by the generators. This aligns nicely with our intuition of what a map between complexes should look like: we tell where the geometry goes ("this edge goes there"), and the algebra is "dragged along for the ride". This gives us the diagram:
    G3--∂-→G2--∂-→G1
|      |      |
f3     f2     f1
↓      ↓      ↓
0 →H3--∂-→H2--∂-→H1

where the fi are homomorphisms . So, this means we can talk about kernels and images!
    Ker(f3)----→Ker(f2)--→Ker(f1)
|         |          |
↓         ↓          ↓
G3--∂----→G2----∂---→G1--→ 0
|         |          |
f3        f2         f1
↓         ↓          ↓
Im(f3)--∂--→Im(f2)--∂-→Im(f1)

F → E → V → {0}
{0} → {e} → {v} → {0}

The Snake Lemma gives us a mapping $d: Ker(f1) \rightarrow Im(f3)$ such that this long exact sequence is saatisfied:

### § What do we wish to compute?

• Now that we've agreed that this family of maps $\{ f_i : G_i \rightarrow H_i \}$ought to be structured maps, the next question is "OK, now what? What does one want to determine"? Ideally, we would get a new chain complex which I tacitly denote as $\{ f(G_i) \}$, consisting of the image of $G_i$ inside $H_i$ and the ability to determine its structure.
• However, this is the boring bit. We don't really care about the chain complex $\{ f(G_i) \}$ per se. What we actually care about are the homology groups! So we would really like a tool that allows us to compute $H_i(f(G))$ in some convenient fashion.