**Surgery equivalence relations on homology cylinders and the core of the Casson invariant**

Prof. Dr. Gwénaël Massuyeau's talk

Date: 25.05.11 Time: 11.20 - 12.10 Room: Gwatt Zentrum

Coauthor: Jean-Baptiste Meilhan

**Abstract:**Let S be a compact oriented surface with one boundary component. Homology cylinders over S form a monoid IC(S) into which the Torelli group I(S) of S 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 E of M with a homeomorphism belonging to the k-th term of the lower central series of the Torelli group of E. The "J_k-equivalence" relation on IC(S) is defined in a similar way using the k-th term of the Johnson filtration. In this talk, we shall review what is known about these equivalence relations. Next, using the LMO homomorphism, we will characterize the Y_3-equivalence and the J_3-equivalence in terms of a few classical invariants. If time allows, we will also show that Morita's "core" of the Casson invariant (which is originally defined on the Johnson subgroup) has a unique extension to (the corresponding submonoid of) IC(S) that is preserved by Y_3-equivalence and the mapping class group action.