The $A_{\infty}$-coalgebra Structure on the $\mathbb{Z}_2$-homology of Closed Compact Surfaces

Let $X$ be a closed compact surface. If $X$ is orientable or the real projective plane, the higher order $A_{\infty}$-coalgebra structure on the $\mathbb{Z}_{2} $-homology of $X$ is degenerate. However, if $X$ is non-orientable of genus $g\geq2,\ H_{\ast}\left( X;\mathbb{Z}_{2}\right) $ admits a non-trivial $A_{\infty}$-colagebra structure whose $k$-ary structure operation $\Delta_{k}$ is non-trivial if and only if $2\leq k<2g$.