What is the definition of the homology group with real coefficients? Consider a surface $S_g$ of genus $g$. The first homology group $H_1(S, \mathbb{Z})$ is the abelianization of $\pi_1(S)$, and is easy to vizualize (we get two generators for each handle).
How is $H_1(S, \mathbb{R})$ defined? Is it the same as $H_1(S, \mathbb{Z})$ but with real coefficients? How does one interpret a 1-cycle with, say a $\sqrt{2}$ coefficient?

If $X$ is a space and $S(X)$ is the chain complex of its singular simplices, then the homology with integer coefficients is simply the homology of $S(X)$.
Now if $A$ is an abelian group, the homology of $X$ with coefficients in $A$ is simply the homology of the complex $A\otimes S(X)$, which is exactly the chain complex with elements the formal linear combinations of singular simplices with coefficients in $A$.
