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 doesone want to determine"? Ideally, we would get a
*new* chain complex whichI 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 toolthat allows us to compute $H_i(f(G))$ in some convenient fashion.