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The extension of Johnson’s homomorphism from the Torelli group to the mapping class group. (English) Zbl 0787.57008
The mapping class group \(M_ g\) of a Riemannian surface \(\Sigma_ g\) of genus \(g\geq 2\) is studied through its action on \(\pi_ 1(\Sigma_ g)\). The author represents \(M_ g\) by its action on the lower central series of \(\pi_ 1(\Sigma_ g)\) using methods of Dennis Sullivan and D. Johnson. The main theorems show that Johnson’s homomorphism can be extended from the Torelli group to the mapping class group via a crossed homomorphism.

MSC:
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57R50 Differential topological aspects of diffeomorphisms
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