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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