Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 16, Number 2 (2016), 899-943.
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 –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.
Algebr. Geom. Topol., Volume 16, Number 2 (2016), 899-943.
Received: 30 December 2014
Revised: 20 April 2015
Accepted: 6 May 2015
First available in Project Euclid: 16 November 2017
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Song, Minkyoung. Invariants and structures of the homology cobordism group of homology cylinders. Algebr. Geom. Topol. 16 (2016), no. 2, 899--943. doi:10.2140/agt.2016.16.899. https://projecteuclid.org/euclid.agt/1510841156