Massey products in the cohomology of groups with applications to link theory
Author:
David Stein
Journal:
Trans. Amer. Math. Soc. 318 (1990), 301325
MSC:
Primary 57M25; Secondary 55S30
MathSciNet review:
958903
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Abstract: Invariants of links in are developed using a modification of the Massey product of onedimensional classes in the cohomology of certain groups. The theory yields two types of invariants, invariants which depend upon a collection of meridians, or basing, of a link, and invariants which do not. The invariants, which are independent of the basing, are compared with John Milnor's invariants. For two component links, a collection of ostensibly based invariants is shown to be independent of the basing. If the linking number of the components of such a link is zero, the resulting invariants may be equivalent to the SatoLevineCochran invariants.
 [1]
Tim
D. Cochran, Geometric invariants of link cobordism, Comment.
Math. Helv. 60 (1985), no. 2, 291–311. MR 800009
(87f:57021), http://dx.doi.org/10.1007/BF02567416
 [2]
, Derivatives of links: Milnor's concordance invariants and Massey's products, preprint.
 [3]
William
G. Dwyer, Homology, Massey products and maps between groups,
J. Pure Appl. Algebra 6 (1975), no. 2, 177–190.
MR
0385851 (52 #6710)
 [4]
Roger
A. Fenn, Techniques of geometric topology, London Mathematical
Society Lecture Note Series, vol. 57, Cambridge University Press,
Cambridge, 1983. MR 787801
(87a:57002)
 [5]
David
Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431–449. MR 0202136
(34 #2010), http://dx.doi.org/10.1090/S00029947196602021361
 [6]
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 (40 #8039)
 [7]
W.
S. Massey, Some higher order cohomology operations, Symposium
internacional de topología algebraica International symposium on
algebraic topology, Universidad Nacional Autónoma de México
and UNESCO, Mexico City, 1958, pp. 145–154. MR 0098366
(20 #4826)
 [8]
John
Milnor, Link groups, Ann. of Math. (2) 59
(1954), 177–195. MR 0071020
(17,70e)
 [9]
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
(19,1070c)
 [10]
Richard
Porter, Milnor’s 𝜇invariants
and Massey products, Trans. Amer. Math.
Soc. 257 (1980), no. 1, 39–71. MR 549154
(81a:57021), http://dx.doi.org/10.1090/S00029947198005491549
 [11]
Dale
Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley,
Calif., 1976. Mathematics Lecture Series, No. 7. MR 0515288
(58 #24236)
 [12]
N. A. Sato, Cobordism of semiboundary links, Topology Appl. 18 (1984).
 [13]
John
Stallings, Homology and central series of groups, J. Algebra
2 (1965), 170–181. MR 0175956
(31 #232)
 [14]
D. W. Stein, Massey products in the cohomology of groups with applications to link theory, Ph.D. Thesis, Brandeis Univ., 1986.
 [15]
, Computing Massey product invariants of links, Proc. 1987 Georgia Topology Conference, Topology Appl. (to appear).
 [16]
V.
G. Turaev, The Milnor invariants and Massey products, Zap.
Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)
66 (1976), 189–203, 209–210 (Russian, with
English summary). Studies in topology, II. MR 0451251
(56 #9538)
 [1]
 T. D. Cochran, Geometric invariants of link cobordism, Comment. Math. Helv. 60 (1985), 291311. MR 800009 (87f:57021)
 [2]
 , Derivatives of links: Milnor's concordance invariants and Massey's products, preprint.
 [3]
 W. G. Dwyer, Homology, Massey products, and maps between groups, J. Pure and Appl. Algebra 6 (1975), 177190. MR 0385851 (52:6710)
 [4]
 R. Fenn, Techniques of geometric topology, London Math. Soc. Stud. Text, No. 57, Cambridge Univ. Press, Cambridge, 1983. MR 787801 (87a:57002)
 [5]
 D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431449. MR 0202136 (34:2010)
 [6]
 W. S. Massey, Higher order linking numbers, Conf. on Algebraic Topology (Univ. of Illinois at Chicago Circle, 1968), Univ. of Chicago at Chicago Circle, Chicago, Ill., pp. 174205. MR 0254832 (40:8039)
 [7]
 , Some higher order cohomology operations, Symposium International de Topologia Algebraica, 1958, pp. 145154. MR 0098366 (20:4826)
 [8]
 J. Milnor, Link groups, Ann. of Math. (2) 2 (1954), 177195. MR 0071020 (17:70e)
 [9]
 , Isotopy of links, Algebraic Geometry and Topology: A Symposium in Honor of Solomon Lefshetz, Princeton Univ. Press, Princeton, N.J., 1957, pp. 280306. MR 0092150 (19:1070c)
 [10]
 R. Porter, Milnor's invariants and Massey products, Trans. Amer. Math. Soc. 257 (1980), 3971. MR 549154 (81a:57021)
 [11]
 D. Rolfsen, Knots and links, Publish or Perish, Berkeley, California, 1976. MR 0515288 (58:24236)
 [12]
 N. A. Sato, Cobordism of semiboundary links, Topology Appl. 18 (1984).
 [13]
 J. Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170181. MR 0175956 (31:232)
 [14]
 D. W. Stein, Massey products in the cohomology of groups with applications to link theory, Ph.D. Thesis, Brandeis Univ., 1986.
 [15]
 , Computing Massey product invariants of links, Proc. 1987 Georgia Topology Conference, Topology Appl. (to appear).
 [16]
 V. G. Turaev, The Milnor invariants and Massey products, Studies in Topology. II, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Kogo oi Stet. Acad. Nauk USSR 66 (1976) (translated in J. Soviet Math.). MR 0451251 (56:9538)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199009589033
PII:
S 00029947(1990)09589033
Article copyright:
© Copyright 1990
American Mathematical Society
