  Among its many corollaries, Poincare duality implies that the de Rham
cohomology of a compact oriented manifold is a shifted commutative Frobenius
algebra --- a commutative Frobenius algebra in which the comultiplication has
cohomological degree equal to the dimension of the manifold. We study the
question of whether this structure lifts to a "homotopy" shifted commutative
Frobenius algebra structure at the cochain level. To make this question
nontrivial, we impose a mild locality-type condition that we call
"quasilocality": strict locality at the cochain level is unreasonable, but it
is reasonable to ask for homotopically-constant families of operations that
become local "in the limit." To make the question concrete, we take the
manifold to be the one-dimensional circle.
  The answer to whether a quasilocal homotopy-Frobenius algebra structure
exists 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: one can work at "tree
level," corresponding roughly to the world of operadic homotopy algebras and
their homotopy modules; or one can work at "graph level," corresponding to the
world of PROPs. For the tree-level version of our question, the answer is the
unsurprising "Yes, such a structure exists" --- indeed, it is unique up to a
contractible space of choices. But for the graph-level version, the answer is
the surprising "No, such a structure does not exist." Most of the paper
consists of computing explicitly this nonexistence, which is controlled by the
numerical value of a certain obstruction, and we compute this value explicitly
via a sequence of integrals.
