Extending coboundary to cocycle I was reading these notes and got confused at the beginning of section 9.2 (p. 21-22).
$i^n$ is the map from $Z^n$ (i.e. ($\ker\partial^n$))to $B^n$ (i.e. $\text{im}(\partial^{n-1})$) of a cochain complex
$$...\to C^n\xrightarrow{\partial^n}C^{n+1}\xrightarrow{\partial^{n+1}}C^{n+2}\to ...$$
From the long exact sequence in homology
$$...\to Z^{n-1}\xrightarrow{i^{n-1}} B^{n-1}\to H^n(C_{\bullet})\to Z^n\xrightarrow{i^{n}} ...$$
we get the short exact sequence
$$0\to \text{coker}(i^{n-1})\to B^{n-1}\to H^n(C_{\bullet})\to \ker(i^n)\to 0$$
Thus far, is absolutely clear.

If we are working over a field and assuming that $H^n(C_{\bullet})$ is finite dimensional, $\text{coker}(i^{n-1}) = 0$ as every $f \in B^{n−1}$
  can be extended to $\tilde{f}\in Z^{n−1}$.

This, on the other hand, is completely unclear to me. 
As far as I understand, elements of $B^{n-1}$ are elements of $\text{im}(\partial^{n-2})$, that is, maps $f$ of the form
$$\text{im}(f:C_{n-1}\xrightarrow{\partial_{n-1}}C_{n-2}\xrightarrow{\sigma}G)$$
while those in $Z^{n-1}$ are of the form
$$\ker(\tilde{f}:C_{n}\xrightarrow{\partial_{n}}C_{n-1}\xrightarrow{\sigma}G)$$
What does it mean to extend $f$ to $\tilde{f}$, and why do we need $H^n(C_{\bullet})$ to be finite-dimensional?
 A: Your confusion stems from the fact that you are using the wrong definition of $Z^n$ and $B^n$.  They are not defined to be $\ker(\partial^n)$ and $\operatorname{im}(\partial^{n-1})$.  Rather, they are defined to be the duals of $Z_n$ and $B_n$; that is, $Z^n=\operatorname{Hom}(Z_n,G)$ and $B^n=\operatorname{Hom}(B_n,G)$.  The map $i^n$ is then the map that takes an element of $Z^n$ and restricts it to the subgroup $B_n$ of $Z_n$ to get an element of $B^n$.  So to show that $i^n$ is surjective, it does make sense to talk about extending elements of $B^n$ to elements of $Z^n$: you are taking a homomorphism $B_n\to G$ and trying to extend it to a homomorphism $Z_n\to G$.  If you are working with vector spaces, this is always possible: a linear map on a subspace of a vector space always extends to the entire vector space (since you can take a basis for the subspace and extend it to a basis for the entire space).
(The assumption that $H^n(C_\bullet)$ is finite-dimensional is totally irrelevant, and I don't know why they mentioned it.)
