Invariants and structures of the homology cobordism group of homology cylinders

The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor’s link invariants and Johnson homomorphisms. They give rise to a descending filtration of the homology cobordism group of homology cylinders. We show that each successive quotient of the filtration is free abelian of finite rank. Second, we define Hirzebruch-type intersection form defect invariants obtained from iterated $p$ –covers for homology cylinders. Using them, we show that the abelianization of the intersection of our filtration is of infinite rank. Also we investigate further structures in the homology cobordism group of homology cylinders which previously known invariants do not detect.

Keywords

homology cylinder, homology cobordism, Milnor invariant, Hirzebruch-type invariant

Mathematical Subject Classification 2010

Primary: 57M27, 57N10

References

Publication

Received: 30 December 2014

Revised: 20 April 2015

Accepted: 6 May 2015

Published: 26 April 2016

Authors

Minkyoung Song
Department of Mathematics POSTECH Pohang 790–784 South Korea

Volume 16, issue 2 (2016)

Minkyoung Song

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors