  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.
