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Let R be a compact oriented surface of genus g with one boundary component.Homology cylinders over R form a monoid IC into which the Torelli group I of Rembeds by the mapping cylinder construction. Two homology cylinders M and M'are said to be Y_k-equivalent if M' is obtained from M by "twisting" anarbitrary surface S in M with a homeomorphim belonging to the k-th term of thelower central series of the Torelli group of S. The J_k-equivalence relation onIC is defined in a similar way using the k-th term of the Johnson filtration.In this paper, we characterize the Y_3-equivalence with three classicalinvariants: (1) the action on the third nilpotent quotient of the fundamentalgroup of R, (2) the quadratic part of the relative Alexander polynomial, and(3) a by-product of the Casson invariant. Similarly, we show that theJ_3-equivalence is classified by (1) and (2). We also prove that the core ofthe Casson invariant (originally defined by Morita on the second term of theJohnson filtration of I) has a unique extension (to the corresponding submonoidof IC) that is preserved by Y_3-equivalence and the mapping class group action.

Equivalence relations for homology cylinders and the core of the Casson invariant