  Let $M$ be a closed, oriented manifold. We prove that the quasi-isomorphism
class of the $Frob_\infty^0$-bialgebra structure on $H^*(M)$ induced by the
open TFT on $\Omega^*(M)$ is a homotopy invariant of the manifold. This is a
three step process. First, we describe the $Frob_\infty^0$-bialgebra on
$H^*(M)$ induced by the partial $Frob_\infty^0$-bialgebra on $\Omega^*(M)$. We
then describe the $Frob_\infty^0$-bialgebra on $H^*(M)$ induced by the cyclic
$C_\infty$-algebra on $H^*(M)$. Finally, we show these two
$Frob_\infty^0$-bialgebras are the same. Since the cyclic $C_\infty$-algebra is
a homotopy invariant, this proves our claim.
