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

Abstract

Let $\Sigma$ be a compact oriented surface of genus $g$ with one boundary component. Homology cylinders over $\Sigma$ form a monoid $\mathcal {IC}$ into which the Torelli group $\mathcal {I}$ of $\Sigma$ 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 $S\subset M$ 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 $\mathcal {IC}$ 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 $\Sigma$, (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 $\mathcal {I}$) has a unique extension (to the corresponding submonoid of $\mathcal {IC}$) that is preserved by $Y_3$-equivalence and the mapping class group action.