Equivalence relations for homology cylinders and the core of the Casson invariant
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_kequivalent if M' is obtained from M by "twisting" an arbitrary surface S in M with a homeomorphim belonging to the kth term of the lower central series of the Torelli group of S. The J_kequivalence relation on IC is defined in a similar way using the kth term of the Johnson filtration. In this paper, we characterize the Y_3equivalence 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 byproduct of the Casson invariant. Similarly, we show that the J_3equivalence 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_3equivalence and the mapping class group action.
 Publication:

arXiv eprints
 Pub Date:
 April 2011
 DOI:
 10.48550/arXiv.1104.2763
 arXiv:
 arXiv:1104.2763
 Bibcode:
 2011arXiv1104.2763M
 Keywords:

 Mathematics  Geometric Topology;
 57M27;
 57N10;
 20F38
 EPrint:
 63 pages. One reference added and some minor modifications in this final version