Q:

Dimension of the homology group with coefficients in $\mathbb{Z}/2\mathbb{Z}$

I asked this on math.stackexchange.com, but didn't get a single answer.
Charles Weibel writes in his survey of homological algebra

Riemann defined a surface $S$ to be $(n + 1)$-fold
  connected if there exists a family $C$ of $n$ closed curves $C_j$ on $S$ such that no subset
  of $C$ forms the complete boundary of a part of $S$, and $C$ is maximal with this
  property. For example, $S$ is simply connected (in the modern sense) if it is $1$-fold
  connected. As we shall see, the connectness number of $S$ is the homology invariant
  $$1 + dim(H_1(S;\mathbb{Z}/2\mathbb{Z})).$$

Can someone explain to me, why he has to take coefficients in $\mathbb{Z}/2\mathbb{Z}$? Shouldn't it be just the first betti number of $S$? If not, why?

A:

$\mathbb{Z}/2\mathbb{Z}$ is easier to handle than $\mathbb{Z}$. At least it is a field. Rational coefficients are not good: for example, consider the projective plane. 

A:

One way this formula is proved is using intersection number of oriented curves. The proof works just fine on any orientable surface using coefficients in any field, both for defining Betti numbers and for defining intersection number. 
On a nonorientable surface, intersection number with coefficients in a field of characteristic $\ne 2$ is not defined (the deeper reason being failure of Poincare duality). But when you plug in a coefficient field of characteristic 2, poof, it all works.

