Invariants and structures of the homology cobordism group of homology cylinders

Song, Minkyoung¹

¹Department of Mathematics, POSTECH, Pohang 790–784, South Korea

Algebraic and Geometric Topology, Tome 16 (2016) no. 2, pp. 899-943

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.899

Classification : 57M27, 57N10
Keywords: homology cylinder, homology cobordism, Milnor invariant, Hirzebruch-type invariant

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