Tree- versus graph-level quasilocal Poincare duality on S^1

TL;DR

This paper investigates whether quasilocal homotopy Frobenius algebra structures exist on the cochains of the circle, revealing that such structures exist at the tree level but not at the graph level, with explicit computations of obstructions.

Contribution

It demonstrates the nonexistence of quasilocal homotopy Frobenius structures at the graph level and provides explicit calculations of the obstruction controlling this nonexistence.

Findings

Tree-level quasilocal homotopy Frobenius structures exist and are unique.

Graph-level structures do not exist due to a nontrivial obstruction.

Explicit integral computations determine the obstruction value.

Abstract

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.