Homotopy Inner Products for Cyclic Operads

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 $\hat{\mathcal{O}}$, which describes modules over $\mathcal{O}$ with invariant inner products. We show that $\hat{\mathcal{O}}$ satisfies Koszulness and identify algebras over a resolution of $\hat{\mathcal{O}}$ 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 Poincaré duality space.

To be published in JHRS