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

TL;DR

This paper characterizes certain equivalence relations among homology cylinders using classical invariants and extends the core of the Casson invariant uniquely within this framework.

Contribution

It provides a complete classification of Y_3- and J_3-equivalence relations for homology cylinders using classical invariants and extends the Casson invariant's core uniquely.

Findings

Y_3-equivalence characterized by three classical invariants.

J_3-equivalence classified by two invariants.

Unique extension of the Casson invariant's core preserved by Y_3-equivalence.

Abstract

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 R 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 in 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 Y_3-equivalence with three classical invariants: (1) the action on the third nilpotent quotient of the fundamental group of R, (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.