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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Invariants and structures of the homology cobordism group of homology cylinders

Minkyoung Song

Algebraic & Geometric Topology 16 (2016) 899–943

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.

homology cylinder, homology cobordism, Milnor invariant, Hirzebruch-type invariant
Mathematical Subject Classification 2010
Primary: 57M27, 57N10
Received: 30 December 2014
Revised: 20 April 2015
Accepted: 6 May 2015
Published: 26 April 2016
Minkyoung Song
Department of Mathematics
Pohang 790–784
South Korea