Homotopy Inner Products for Cyclic Operads

TL;DR

This paper generalizes the concept of homotopy inner products to all cyclic quadratic Koszul operads, introducing a new colored operad framework and demonstrating its applications to Poincaré duality spaces.

Contribution

It defines a new colored operad hat{O} for modules with invariant inner products, proves its Koszulness, and constructs homotopy inner products for cochains of Poincare9 duality spaces.

Findings

hat{O} is Koszul.

Algebras over a resolution of hat{O} are described by derivations and module maps.

Constructed homotopy inner products on cochains of Poincare9 duality spaces.

Abstract

We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\widehat{\mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\widehat{\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\widehat{\mathcal O}$ in terms of derivations and module maps. As an application we construct a homotopy inner product over the commutative operad on the cochains of any Poincar\'e duality space.