An A ∞-coalgebra structure on a closed compact surface

Let P be an n-gon with n >= 3. There is a formal combinatorial A(infinity)-coalgebra structure on cellular chains C-*(P) with non-vanishing higher order structure when n >= 5. If X-g is a closed compact surface of genus g >= 2 and P-g is a polygonal decomposition, the quotient map q : P-g -> Xg projects the formal A(infinity)-coalgebra structure on C-*(P-g) to a quotient structure on C-*(X-g), which persists to homology H-*(X-g; Z(2)), 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 with g >= 2 or unorientable with g >= 3. But whether or not the A(infinity)-coalgebra structure on homology observed here is topologically invariant is an open question.(1)