COHOMOLOGICAL ASPECTS OF MAGNUS EXPANSIONS.

By Nariya Kawazumi. 2005

125 citations recorded by the source corpus.

We generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group of a free group Aut(Fn).The extended ones are not homomor- phisms, but satisfy an infinite sequence of coboundary relations, so that we call them the Johnson maps.In this paper we confine ourselves to studying the first and the second relations, which have cohomological consequences about the group Aut(Fn) and the mapping class groups for surfaces.The first one means that the first Johnson map is a twisted 1-cocycle of the group Aut(Fn).Its cohomology class coincides with "the unique elementary particle" of all the Morita-Mumford classes on the mapping class group for a surface (Ka1) (KM1).The second one restricted to the mapping class group is equal to a fundamental relation among twisted Morita-Mumford classes pro- posed by Garoufalidis and Nakamura (GN) and established by Morita and the author (KM2).This means we give a coherent proof of the fundamental relation.The first Johnson map gives the abelianization of the induced automorphism group IAn of a free group in an explicit way.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/26ec0100d343abd6f16ab5b08a61338f21c50520

How researchers see this paper

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In Geometry and Topology

1. Y. Kuno
2. G. Massuyeau
3. Alexandra Pettet
4. Frederick R. Cohen
5. Nariya Kawazumi
6. Jie Wu
7. Jonathan Pakianathan
8. V. Vershinin

In Mathematical Physics

1. Y. Kuno
2. G. Massuyeau
3. Alexandra Pettet
4. Frederick R. Cohen
5. Nariya Kawazumi
6. Jie Wu
7. Jonathan Pakianathan
8. V. Vershinin

Citations in the corpus

• The logarithms of Dehn twists (2010, methodology): “Another approach by using the notion of generalized Magnus expansions is developed in [14].”
• Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant (2008, methodology): “Expansions have been studied by Lin in the context of Vassiliev invariants and Milnor’s μ invariants [28], and by Kawazumi in connection with Johnson homomorphisms and the cohomology of mapping class groups [21].”
• Representation theory and homological stability (2010, unknown): “[Ka]) gives: H1(IAn;Q) ≈ ∧2 Qn ⊗ (Qn)∗ ≈ V (L1 + L2 − Ln)⊕ V (L1) (20)”
• The Johnson homomorphism and the second cohomology of IAn (2005, methodology): “: IA(2) −→ ∧2H 1(F) ⊗H1(F)/∧3H1(F) ⊗H1(F)∗ which is analogous to the Johnson homomorphism for IA. A generalization of this map to all IA(n) and to some related classes of groups is discussed in [CP], [K], and [Sat]. We begin with a map δ(2): IA(2) −→ Hom(H 1(F),F(3)/F(4)) defined as follows: Algebrai”
• Basis-conjugating automorphisms of a free group and associated Lie algebras (2006, background): “20F28; 20F40, 20J06”