The loop product for 3-manifolds (Q1876835)

Let \(LM\) be the free loop space of a \(d\)-dimensional connected oriented smooth closed manifold \(M\). As proved by Chas and Sullivan the graded abelian group \(H_{*-d}(LM)\) equipped with the loop product is a graded commutative algebra. Moreover, the loop product restricts to the intersection product via the injection \(H_*(M)\hookrightarrow H_*(LM)\) induced by the natural section. In fact we have a natural splitting \(H_* (LM)=H_*(M)\oplus A_M\) and the author characterizes the closed 3-manifolds whose loop product restricted to \(A_M\) is trivial: all the finite covers of any 3-manifold \(M\) with contractible universal cover and such that \(\pi_1(M)\) has no rank 2 abelian subgroup.