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A Free Frobenius Bialgebra Structure of Differential Forms

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 $Ω^*(M)$ 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 $Ω^*(M)$. 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.