Two 3-manifolds are called Y_k-equivalent if one can be obtained from the other by "twisting" an embedded surface by an element of the k-th term of the lower central series of its Torelli group. The J_k-equivalence relation is defined similarly, using the Johnson filtration instead of the lower central series. In this talk, we shall consider these equivalence relation among homology cylinders over a given surface S, which are 3-manifolds homologically equivalent to S \times [0,1]. We classify these equivalence relations, for k \le 3, using several classical invariants. This provides generalizations of results of W.Pitsch and S.Morita on the structure of integral homology spheres and the Casson invariant.
This is a joint work with G. Massuyeau.