For Σ a compact connected oriented surface, we consider homology cylinders over Σ: these are homology cobordisms with an extra homological triviality condition. When considered up to Y 2 -equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian group. In this paper, when Σ has one or zero boundary component, we define a surgery map from a certain space of graphs to this group. This map is shown to be an isomorphism, with inverse given by some extensions of the first Johnson homomorphism and Birman-Craggs homomorphisms.
Abstract. Let Σ be a compact oriented surface of genus g with one boundary component. Homology cylinders over Σ form a monoid IC into which the Torelli group I of Σ embeds 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" an arbitrary surface S ⊂ M with a homeomorphim belonging to the k-th term of the lower central series of the Torelli group of S. The J k -equivalence relation on IC is defined in a similar way using the k-th term of the Johnson filtration. In this paper, we characterize the Y3-equivalence with three classical invariants: (1) the action on the third nilpotent quotient of the fundamental group of Σ, (2) the quadratic part of the relative Alexander polynomial, and (3) a by-product of the Casson invariant. Similarly, we show that the J3-equivalence is classified by (1) and (2). We also prove that the core of the Casson invariant (originally defined by Morita on the second term of the Johnson filtration of I) has a unique extension (to the corresponding submonoid of IC) that is preserved by Y3-equivalence and the mapping class group action.
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