Maurer-Cartan Elements and Cyclic Operads

First we argue that many BV and homotopy BV structures, including both familiar and new examples, arise from a common underlying construction. The input of this construction is a cyclic operad along with a cyclically invariant Maurer-Cartan element in an associated Lie algebra. Using this result we introduce and study the operad of cyclically invariant operations, with instances arising in cyclic cohomology and $S^1$ equivariant homology. We compute the homology of the cyclically invariant operations; the result being the homology operad of $\mathcal{M}_{0,n+1}$, the uncompactified moduli spaces of punctured Riemann spheres, which we call the gravity operad after Getzler. Motivated by the line of inquiry of Deligne's conjecture we construct `cyclic brace operations' inducing the gravity relations up-to-homotopy on the cochain level. Motivated by string topology, we show such a gravity-BV pair is related by a long exact sequence. Examples and implications are discussed in course.