A Batalin–Vilkovisky algebra structure on the Hochschild cohomology of truncated polynomials

We calculate the Batalin–Vilkovisky, and the induced Gerstenhaber structures of the Hochschild cohomology of the singular R-cochain complex of n-dimensional K-projective spaces, where K = C , H and R = Z and any field. In the special case that K = C , n = 1 and R = Z , we show that this structure cannot be identified with the BV-structure of the integral loop homology of the 2-dimensional sphere computed by L. Menichi, but their induced Gerstenhaber structures can still be identified. Combined with the work of Y. Félix and J. Thomas, the main result of the present paper calculates the BV-structure of the rational loop homology of projective spaces.