On String Topology of Three Manifolds

Let M be a closed, oriented and smooth manifold of dimension d. Let LM be the space of smooth loops inM . In [5] Chas and Sullivan introduced loop product, a product of degree −d on the homology of LM . In this paper we show how for three manifolds the “nontriviality” of the loop product relates to the “hyperbolicity” of the underlying manifold. This is an application of the existing powerful tool and results in three dimensional topology such as the prime decomposition, torus decomposition, Seifert theorem, torus theorem.