Morita's trace maps on the group of homology cobordisms.

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 15 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. [ABSTRACT FROM AUTHOR]