EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 6 June 2010. For the current production of this journal, please refer to http://www.springer.com/mathematics/geometry/journal/40062.

Homotopy Inner Products for Cyclic Operads

Homotopy Inner Products for Cyclic Operads

Riccardo Longoni and Thomas Tradler

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

Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 343-358