On string topology of three manifolds (Q2488281)

Let \(L(X)\) be the free loop space on \(X\). In their seminal preprint, [math.GT/9911159], \textit{M. Chas} and \textit{D. Sullivan} proved that for an oriented smooth \(d\)-manifold the desuspended homology of \(M\), \(H_{*-d}(LM)\) is a graded commutative algebra. Here the multiplication is called the loop-product. Moreover, the canonical projection \(LM\to M\), \(f\mapsto f(0)= f(1)\) induces a homomorphism of graded algebras \(p: H_{*-d}(LM)\to H_{*-d}(M)\). Here the multiplication on \(H_{*-d}(M)\) is the intersection product. The following is the main result of this paper. Assume that \(\dim M= 3\). If \(M\) is aspherical and its fundamental group has no rank 2 abelian subgroup then the restriction of the loop-product on the ideal \(\text{Ker\,}p\) is trivial for \(M\) and all its finite covers. Otherwise, the restriction of the loop-product on the ideal \(\text{Ker\,}p\) is nontrivial for \(M\) or a double cover of \(M\). Although the main result as stated above concerns the closed 3-manifolds, throughout the paper the author identifies several classes of 3-manifolds with boundary where the restriction of the loop-product on the ideal \(\text{Ker\,}p\) is nontrivial. The proof of these results rely on some existing powerful tools in three-dimensional topology.