## Algebraic & Geometric Topology

### Invariants and structures of the homology cobordism group of homology cylinders

Minkyoung Song

#### Abstract

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 2 (2016), 899-943.

Dates
Revised: 20 April 2015
Accepted: 6 May 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841156

Digital Object Identifier
doi:10.2140/agt.2016.16.899

Mathematical Reviews number (MathSciNet)
MR3493411

Zentralblatt MATH identifier
1352.57024

#### Citation

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

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