Chains(R) does not admit a geometrically meaningful properadic homotopy Frobenius algebra structure
arXiv:1308.3423
Abstract
The embedding Chains(R) into Cochains(R) as the compactly supported cochains might lead one to expect Chains(R) to carry a nonunital commutative Frobenius algebra structure, up to a degree shift and some homotopic weakening of the axioms. We prove that under reasonable "locality" conditions, a cofibrant resolution of the dioperad controlling nonunital shifted-Frobenius algebras does act on Chains(R), and in a homotopically-unique way. But we prove that this action does not extend to a homotopy Frobenius action at the level of properads or props. This gives an example of a geometrically meaningful algebraic structure on homology that does not lift in a geometrically meaningful way to the chain level.
15 pages, diagrams in TikZ. An expanded retelling of the story in this paper, with different conventions and clearer results, is available in arXiv:1412.4664