EQUIVALENCE RELATIONS FOR HOMOLOGY CYLINDERS AND THE CORE OF THE CASSON INVARIANT

作者：Massuyeau Gwenael^*; Meilhan Jean Baptiste

来源：Transactions of the American Mathematical Society, 2013, 365(10): 5431-5502.

DOI：10.1090/S0002-9947-2013-05818-7

Let S be a compact oriented surface of genus g with one boundary component. Homology cylinders over S form a monoid IC into which the Torelli group I of Sigma embeds by the mapping cylinder construction. Two homology cylinders M and M-I are said to be Y-k-equivalent if M%26apos; is obtained from M by %26quot;twisting%26quot; an arbitrary surface S. M with a homeomorphism 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 Y-3-equivalence with three classical invariants: (1) the action on the third nilpotent quotient of the fundamental group of S, (2) the quadratic part of the relative Alexander polynomial, and (3) a by-product of the Casson invariant. Similarly, we show that the J(3)-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 Y-3-equivalence and the mapping class group action.

• 出版日期2013-10