A Free Frobenius Bialgebra Structure of Differential Forms

TL;DR

This paper demonstrates that the Frobenius bialgebra structure on the cohomology of a closed, oriented manifold, derived from open topological field theory, is a homotopy invariant, establishing a new link between algebraic structures and topological invariants.

Contribution

It proves that the Frobenius bialgebra structure on cohomology, induced by open TFT, is a homotopy invariant of the manifold, connecting algebraic and topological properties.

Findings

Frob_infinity^0-bialgebra on H^*(M) is a homotopy invariant.

The induced algebraic structures are equivalent via cyclic C_infinity-algebra.

Homotopy invariance of the algebraic structure is established.

Abstract

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.