Morita’s trace maps on the group of homology cobordisms
Morita introduced in 2008 a [Formula: see text]-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His [Formula: see text]-cocycle contains all the “traces” of Johnson homomorphisms which he introduced 15 years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita’s [Formula: see text]-cocycle based on a simple and explicit construction. Our [Formula: see text]-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.