  How hard is it to fill a loop in a Cayley graph with an unoriented surface?
Following a comment of Gromov in "Asymptotic invariants of infinite groups", we
define homological filling functions of groups with coefficients in a group
$R$. Our main theorem is that the coefficients make a difference. That is, for
every $n \geq 1$ and every pair of coefficient groups $A, B \in
\{\mathbb{Z},\mathbb{Q}\} \cup \{\mathbb{Z}/p\mathbb{Z} : p\text{ prime}\}$,
there is a group whose filling functions for $n$-cycles with coefficients in
$A$ and $B$ have different asymptotic behavior.
