Journal of the Mathematical Society of Japan

Cocycles of nilpotent quotients of free groups

Takefumi NOSAKA

Advance publication

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Abstract

We focus on the cohomology of the $k$-th nilpotent quotient of a free group. We describe all the group 2-, 3-cocycles in terms of the Massey product and give expressions for some of the 3-cocycles. We also give simple proofs of some of the results on the Milnor invariant and Johnson–Morita homomorphisms.

Article information

Source
J. Math. Soc. Japan, Advance publication (2019), 14 pages.

Dates
Received: 3 March 2018
Revised: 15 September 2018
First available in Project Euclid: 18 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1552896022

Digital Object Identifier
doi:10.2969/jmsj/79997999

Subjects
Primary: 55S30: Massey products 20F18: Nilpotent groups [See also 20D15] 20J06: Cohomology of groups
Secondary: 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25} 20F14: Derived series, central series, and generalizations

Keywords
nilpotent group higher Massey product group cohomology mapping class group link

Citation

NOSAKA, Takefumi. Cocycles of nilpotent quotients of free groups. J. Math. Soc. Japan, advance publication, 18 March 2019. doi:10.2969/jmsj/79997999. https://projecteuclid.org/euclid.jmsj/1552896022


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References

  • [BC] P. Benito and D. de-la-Concepción, An overview of free nilpotent Lie algebras, Comment. Math. Univ. Carolin, 3 (2014), 325–339.
  • [Bro] K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994.
  • [CFL] K. T. Chen, R. H. Fox and R. C. Lyndon, Free differential calculus IV, the quotient groups of the lower central series, Ann. of Math., 68 (1958), 81–95.
  • [CGO] T. Cochran, A. Gerges and K. Orr, Dehn surgery equivalence relations on three-manifolds, Math. Proc. Cambridge Philos. Soc., 131 (2001), 97–127.
  • [Day] M. B. Day, Extending Johnson's and Morita's homomorphisms to the mapping class group, Algebr. Geom. Topol., 7 (2007), 1297–1326.
  • [FS] R. Fenn and D. Sjerve, Massey products and lower central series of free groups, Canad. J. Math., 39 (1987), 322–337.
  • [GL] S. Garoufalidis and J. Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, from: Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math., 73, Amer. Math. Soc., Providence, RI, 2005, 173–203.
  • [GG] C. K. Gupta and N. D. Gupta, Generalized Magnus embeddings and some applications, Math. Z., 160 (1978), 75–87.
  • [Hall] M. Hall, A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc., 1 (1950), 575–581.
  • [Heap] A. Heap, Bordism invariants of the mapping class group, Topology, 45 (2006), 851–886.
  • [IO] K. Igusa and K. Orr, Links, pictures and the homology of nilpotent groups, Topology, 40 (2001), 1125–1166.
  • [Joh1] D. Johnson, An abelian quotient of the mapping class group $\mathcal{I}_g$, Math. Ann., 249 (1980), 225–242.
  • [Joh2] D. Johnson, A survey of the Torelli group, Contemporary Mathematics, 20, American Mathematical Society, Providence, RI, 1983, 165–179.
  • [Ki] T. Kitano, Johnson's homomorphisms of subgroups of the mapping class group, the Magnus expansion and Massey higher products of mapping tori, Topology Appl., 69 (1996), 165–172.
  • [Kra] D. Kraines, Massey higher products, Trans. Amer. Math. Soc., 124 (1966), 431–449.
  • [KN] H. Kodani and T. Nosaka, Milnor invariants via unipotent Magnus embeddings, preprint.
  • [Mil] J. Milnor, Isotopy of links, In: Algebraic geometry and topology, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, NJ, 1957, 280–306.
  • [MKS] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Interscience Publ., New York, 1966.
  • [Mas] G. Massuyeau, Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant, Bull. Soc. Math. France, 140 (2012), 101–161.
  • [M1] S. Morita, The extension of Johnson's homomorphism from the Torelli group to the mapping class group, Invent. Math., 111 (1993), 197–224.
  • [M2] S. Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. Journal, 70 (1993), 699–726.
  • [O1] K. Orr, Homotopy invariants of links, Inventiones Math., 95 (1989), 379–394.
  • [O2] K. Orr, Link concordance invariants and Massey products, Topology, 30 (1991), 699–710.
  • [Po] R. Porter, Milnor's $\bar{\mu}$-invariants and Massey products, Trans. Amer. Math. Soc., 257 (1980), 39–71.
  • [Tu] V. G. Turaev, Milnor's invariants and Massey products, J. SovietMath., 12 (1979), 128–137.
  • [Witt] E. Witt, Treue Darstellung Liescher Ringe, Journal für die Reine und Angewandte Mathematik, 177 (1937), 152–160.