Charles Weibel writes in his survey of homological algebra
Riemann de fined 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?
