# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] Epstein, D. B. A., Curves on 2-manifolds and isotopies, Acta Math., 115, 83-107, (1966) · Zbl 0136.44605 [2] Garoufalidis, S.; Levine, J., Graphs and patterns in mathematics and theoretical physics, 73, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, 173-203, (2005), Amer. Math. Soc., Providence, RI · Zbl 1086.57013 [3] Goldman, W. M., Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math., 85, 2, 263-302, (1986) · Zbl 0619.58021 [4] Habegger, N., Milnor, Johnson and the tree-level perturbative invariants [5] Jennings, S. A., The group ring of a class of infinite nilpotent groups, Canad. J. Math., 7, 169-187, (1955) · Zbl 0066.01302 [6] Kawazumi, N., Cohomological aspects of Magnus expansions [7] Kawazumi, N.; Kuno, Y., Groupoid-theoretical methods in the mapping class groups of surfaces [8] Kawazumi, N.; Kuno, Y., The logarithms of Dehn twists · Zbl 1361.57027 [9] Kontsevich, M., The Gel’fand Mathematical Seminars, 1990-1992, Formal (non)commutative symplectic geometry, 173-187, (1993), Birkhäuser Boston, Boston, MA · Zbl 0821.58018 [10] Kuno, Y., The generalized Dehn twist along a figure eight · Zbl 1282.57025 [11] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial group theory. Presentations of groups in terms of generators and relations, (1976), Dover Publications, Inc., New York · Zbl 0362.20023 [12] Massuyeau, G., Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant, Bull. Soc. Math. France, 140, 1, 101-161, (2012) · Zbl 1248.57009 [13] Morita, S., Groups of diffeomorphisms, 52, Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces, 443-468, (2008), Math. Soc. Japan, Tokyo · Zbl 1166.57012 [14] Papakyriakopoulos, C. D., Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), 84, Planar regular coverings of orientable closed surfaces, 261-292, (1975), Princeton Univ. Press, Princeton, N.J. · Zbl 0325.55002 [15] Perron, B., A homotopic intersection theory on surfaces: applications to mapping class group and braids, Enseign. Math. (2), 52, 1-2, 159-186, (2006) · Zbl 1161.57009 [16] Quillen, D., Rational homotopy theory, Ann. of Math. (2), 90, 205-295, (1969) · Zbl 0191.53702 [17] Turaev, V. G., Intersections of loops in two-dimensional manifolds, (Russian) Mat. Sb, 106(148), 566-588, (1978) · Zbl 0384.57004 [18] Turaev, V. G., Multiplace generalizations of the Seifert form of a classical knot, (Russian) Mat. Sb, 116(158), 370-397, (1981) · Zbl 0484.57002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.