Homotopy Inner Products for Cyclic Operads

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 $\hat{\mathcal O}$,whichdescribesmodulesover$\mathcal O$ 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. An application to Poincar'e duality on the chain level of a suitable topological space is given.