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Morita’s trace maps on the group of homology cobordisms. (English) Zbl 1485.20097

Summary: S. Morita introduced in 2008 [Adv. Stud. Pure Math. 52, 443–468 (2008; Zbl 1166.57012)] 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.

MSC:

20F38 Other groups related to topology or analysis
20F14 Derived series, central series, and generalizations for groups
20F28 Automorphism groups of groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20J05 Homological methods in group theory
57M05 Fundamental group, presentations, free differential calculus
57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
57K30 General topology of 3-manifolds
57N70 Cobordism and concordance in topological manifolds

Citations:

Zbl 1166.57012
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References:

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