Tree- versus graph-level quasilocal Poincaré duality on S¹

Among its many corollaries, Poincaré duality implies that the de Rham cohomology of a compact oriented manifold is a commutative Frobenius algebra. Focusing on the case of S¹, this paper studies the question of whether this commutative Frobenius algebra structure lifts to a “homotopy” commutative Frobenius algebra structure at the cochain level, under a mild locality-type condition called “quasilocality”. The answer turns out to depend on the choice of context in which to do homotopy algebra—there are two reasonable worlds in which to study structures (like Frobenius algebras) that involve many-to-many operations. If one works at “tree level”, we prove that there is a homotopically-unique quasilocal cochain-level homotopy Frobenius algebra structure lifting the Frobenius algebra structure on cohomology. However, if one works instead at “graph level”, we prove that a quasilocal lift does not exist.