A Free Frobenius Bialgebra Structure of Differential Forms

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