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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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