Morita's trace maps on the group of homology cobordisms

TL;DR

This paper introduces a new explicit version of Morita's 1-cocycle on homology cobordisms, revealing its fundamental properties and applications to the group's abelianization and automorphism filtrations.

Contribution

It proposes a simplified, explicit construction of Morita's 1-cocycle, connecting it with key invariants and demonstrating its non-triviality in the group's abelianization.

Findings

The new 1-cocycle satisfies key algebraic properties.

The rational abelianization of the homology cobordism group is non-trivial.

Comparison of filtrations on automorphism groups using algebraic methods.

Abstract

Morita introduced in 2008 a 1-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His 1-cocycle contains all the "traces" of Johnson homomorphisms which he introduced fifteen years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita's 1-cocycle based on a simple and explicit construction. Our 1-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization of the group of homology cobordisms is non-trivial. Besides, we apply some of our algebraic methods to compare two natural filtrations on the automorphism group of a finitely-generated free group.