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            <title xml:lang="en">Morita's trace maps on the group of homology cobordisms</title>
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                <term xml:lang="en">Morita’s trace</term>
                <term xml:lang="en">Magnus representation</term>
                <term xml:lang="en">Johnson homomorphism</term>
                <term xml:lang="en">Homology cobordism</term>
                <term xml:lang="en">Automorphism group of a free group</term>
                <term xml:lang="en">Mapping class group</term>
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              <p>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.</p>
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