Morita's trace maps on the group of homology cobordisms

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.
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https://hal.archives-ouvertes.fr/hal-01338149
Contributor : Gwénaël Massuyeau <>
Submitted on : Tuesday, June 28, 2016 - 8:46:36 AM
Last modification on : Thursday, December 20, 2018 - 8:46:51 AM

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Gwenael Massuyeau, Takuya Sakasai. Morita's trace maps on the group of homology cobordisms. Journal of Topology and Analysis, 2018, ⟨10.1142/S179352531950064X⟩. ⟨hal-01338149⟩

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