  Let $P$ be an $n$-gon with $n\geq3.$ There is a formal combinatorial
$A_\infty$-coalgebra structure on cellular chains $C_*(P)$ with non-vanishing
higher order structure when $n\geq5$. If $X_g$ is a closed compact surface of
genus $g\geq2$ and $P_g$ is a polygonal decomposition, the quotient map
$q:P_g\to X_g$ projects the formal $A_\infty$-coalgebra structure on $C_*(P_g)$
to a quotient structure on $C_*(X_g)$, which persists to homology
$H_{\ast}\left( X_g;\mathbb{Z}_{2}\right) $, whose operations are determined by
the quotient map $q$, and whose higher order structure is non-trivial if and
only if $X_g$ is orientable or unorientable with $g\geq3$. But whether or not
the $A_{\infty}$-coalgebra structure on homology observed here is topologically
invariant is an open question.
