Milnor’s 𝜇-invariants and Massey products

Porter, Richard

doi:10.1090/S0002-9947-1980-0549154-9

Milnor's $\bar \mu$ -invariants and Massey products

Author: Richard Porter
Journal: Trans. Amer. Math. Soc. 257 (1980), 39-71
MSC: Primary 57Q45; Secondary 55S30
DOI: https://doi.org/10.1090/S0002-9947-1980-0549154-9
MathSciNet review: 549154
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Abstract: The main result of this paper gives an interpretation of Milnor's $\overline \mu$ -invariants of a link in terms of Massey products in the complement of the link. The approach presented here can be used to give topological proofs of results about the $\overline \mu$ -invariants obtained by Milnor using different methods.

[1] K. T. Chen, Commutator calculus and link invariants, Proc. Amer. Math. Soc. 3 (1952), 44–55. MR 46361, https://doi.org/10.1090/S0002-9939-1952-0046361-7
[2] Kuo-Tsai Chen, Isotopy invariants of links, Ann. of Math. (2) 56 (1952), 343–353. MR 49557, https://doi.org/10.2307/1969803
[3] Kuo-tsai Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. (2) 97 (1973), 217–246. MR 380859, https://doi.org/10.2307/1970846
[4] Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 382702, https://doi.org/10.1007/BF01389853
[5] R. Fenn and P. Taylor, Generalized Borromean rings and higher order linking numbers (to appear).
[6] William G. Dwyer, Homology, Massey products and maps between groups, J. Pure Appl. Algebra 6 (1975), no. 2, 177–190. MR 385851, https://doi.org/10.1016/0022-4049(75)90006-7
[7] R. M. Goresky, Geometric cohomology and homology of stratified objects, Ph. D. Thesis, Brown University, 1976.
[8] V. K. A. M. Gugenheim and J. Peter May, On the theory and applications of differential torsion products, American Mathematical Society, Providence, R.I., 1974. Memoirs of the American Mathematical Society, No. 142. MR 0394720
[9] M. A. Gutiérrez, The \overline𝜇-invariants for groups, Proc. Amer. Math. Soc. 55 (1976), no. 2, 293–298. MR 422328, https://doi.org/10.1090/S0002-9939-1976-0422328-8
[10] David Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431–449. MR 202136, https://doi.org/10.1090/S0002-9947-1966-0202136-1
[11] S. Lefschetz, Topology, Amer. Math. Soc. Colloq. Publ., no. 12, Amer. Math. Soc., Providence, R. I., 1930.
[12] Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966. MR 0207802
[13] W. S. Massey, Higher order linking numbers, Conf. on Algebraic Topology (Univ. of Illinois at Chicago Circle, Chicago, Ill., 1968) Univ. of Illinois at Chicago Circle, Chicago, Ill., 1969, pp. 174–205. MR 0254832
[14] J. Peter May, Matric Massey products, J. Algebra 12 (1969), 533–568. MR 238929, https://doi.org/10.1016/0021-8693(69)90027-1
[15] Clint McCrory, Cone complexes and PL transversality, Trans. Amer. Math. Soc. 207 (1975), 269–291. MR 400243, https://doi.org/10.1090/S0002-9947-1975-0400243-7
[16] -, Poincaré duality in spaces with singularities (to appear).
[17] John Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177–195. MR 71020, https://doi.org/10.2307/1969685
[18] John Milnor, Isotopy of links. Algebraic geometry and topology, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957, pp. 280–306. MR 0092150
[19] E. J. O'Neill, Higher order cohomology products and links, Ph. D. Thesis, Yale University, 1976.
[20] Jean-Pierre Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2) 54 (1951), 425–505 (French). MR 45386, https://doi.org/10.2307/1969485
[21] John Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170–181. MR 175956, https://doi.org/10.1016/0021-8693(65)90017-7
[22] John R. Stallings, Quotients of the powers of the augmentation ideal in a group ring, Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Princeton Univ. Press, Princeton, N.J., 1975, pp. 101–118. Ann. of Math. Studies, No. 84. MR 0379685
[23] A. W. Tucker, An abstract approach to manifolds, Ann. of Math. (2) 34 (1933), no. 2, 191–243. MR 1503105, https://doi.org/10.2307/1968201
[24] Hassler Whitney, On products in a complex, Ann. of Math. (2) 39 (1938), no. 2, 397–432. MR 1503416, https://doi.org/10.2307/1968795
[25] S. Wylie, Duality and intersection in general complexes, Proc. London Math. Soc. (2) 46 (1940), 174–198. MR 1904, https://doi.org/10.1112/plms/s2-46.1.174