TITLE: Cubical vs. simplicial singular homology 
QUESTION [78 upvotes]: Singular homology is usually defined via singular simplices, but Serre in his thesis uses singular cubes, which he claims are better adapted to the study of fibre spaces. This young man (25 years old at the time) seemed to know what he was talking about and has had a not too unsuccessful career since.
So my (quite connected) questions are 
1) Why do so few books use this approach ( I only know Massey's) ?
2) What are the pro's and con's of both approaches ?
3) Does it matter ? After all the homology groups obtained are the same.

REPLY [6 votes]: A disadvantage of using cubical singular homology comes form the following example (copied from Natalia Hajlasz's homework):

Example. The circle $c(t)=(\cos 2\pi t,\sin 2\pi t)$, $t\in [0,1]$ is not the
  boundary  of any  singular cubical chain.

Proof. For a singular chain $\sum_i a_i c_i$, where $a_i\in\mathbb{Z}$ and $c_i$ is a singular cube, define its `length' by $\sum_i a_i$. Since opposite sides of a singular cube have signs $\pm 1$, it follows that the length of $\partial c_i$ equals zero and hence
the length of $\partial(\sum_i a_i c_i)$ equals zero. However, the length of  $c(t)=(\cos 2\pi t,\sin 2\pi t)$ equals $1$ so it cannot be represented as a boundary of a singular chain. $\Box$
If you consider the singular cube: $b=(s\cos 2\pi t, s \sin 2\pi t)$, $s,t\in [0,1]$, then $\partial b$ is a difference of two singular cubes: one is $c$ and one is a constant cube (mapping to the center). It is not possible to avoid such degenerate cubes. 
As I understand this problem is what Tyler Lawson had in mind when he wrote in his answer: The main disadvantage that comes solely from the point of view of homology is that you have this irritating normalization procedure: you have to take the cubical chains on X and take the quotient by degenerate cubes.