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Fox pairings and generalized Dehn twists. (Formes de Fox et twists de Dehn généralisés.) (English. French summary) Zbl 1297.57005
Starting with a group and a certain bilinear form on its group algebra (a “Fox pairing”), the main construction of the paper produces a family of group automorphisms of the Malcev completion of the group, generalizing to an algebraic setting the action of the Dehn twists on the group algebras of the fundamental groups of surfaces. This is inspired by work of N. Kawazumi and Y. Kuno which generalized the action of Dehn twists to arbitrary, i.e. not necessarily simple loops on the surface [Groupoid-theoretical methods in the mapping class groups of surfaces; arXiv:1109.6479]. “One simplification achieved here consists in replacing algebra automorphisms of the completed group algebras by group automorphisms of the Malcev completions. The key ingredient in our approach is the homotopy intersection form on surfaces introduced by the second author [Math. USSR, Sb. 35, 229–250 (1979; Zbl 0422.57005)].”

MSC:
57M05 Fundamental group, presentations, free differential calculus
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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