Brackets in the free loop space homology of some homogeneous spaces (Q388747)

Let \(\Bbbk\) be a principal ideal domain and \(M\) be a compact oriented \(d\)-dimensional smooth manifold and denote by \(\mathbb H(M^{S^1}; \Bbbk)\) the homology of the free loop space on \(M\) with coefficients in \(\Bbbk\) with a shift of degrees defined by \(\mathbb H_k(-;\Bbbk)=H_{k+d}(-; \Bbbk) \). Chas and Sullivan (1999) have defined on \(\mathbb H(M^{S^1}; \Bbbk)\) a structure of Batalin-Vilkovisky algebra and in particular, a Gerstenhaber bracket. The purpose of this paper is to compute this bracket in some special cases and for \(\Bbbk\) a field of characteristic zero. The author uses Sullivan minimal models. He also indicates a gap appearing in the proof of a previously published result on the same subject.