Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cobordism Massey products


Author: J. C. Alexander
Journal: Trans. Amer. Math. Soc. 166 (1972), 197-214
MSC: Primary 55G30
DOI: https://doi.org/10.1090/S0002-9947-1972-0293623-1
MathSciNet review: 0293623
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The structure of Massey products is introduced into the bordism ring $ {\Omega ^S}$ of manifolds with structure S and machinery is developed to investigate it. The product is changed to one in homotopy via the Pontrjagin-Thom map and methods for computation via the Adams spectral sequence are developed. To illustrate the methods, some products in $ {\Omega ^{SU}}$ and $ {\Omega ^{Sp}}$ are computed.


References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, On the structure and applications of the Steenrod algebra, Comment. Math. Helv. 32 (1958), 180-214. MR 20 #2711. MR 0096219 (20:2711)
  • [2] -, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20-104. MR 25 #4530. MR 0141119 (25:4530)
  • [3] D. W. Anderson, E. H. Brown, Jr. and F. P. Peterson, SU-cobordism, KO-characteristic numbers, and the Kervaire invariant, Ann. of Math. (2) 83 (1966), 54-67. MR 32 #6470. MR 0189043 (32:6470)
  • [4] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Band 33, Academic Press, New York; Springer-Verlag, Berlin, 1964. MR 31 #750. MR 0176478 (31:750)
  • [5] -, Torsion in SU-bordism, Mem. Amer. Math. Soc. No. 60 (1966). MR 32 #6471. MR 0189044 (32:6471)
  • [6] D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431-449. MR 34 #2010. MR 0202136 (34:2010)
  • [7] A. Liulevicius, Notes on homotopy of Thorn spectra, Amer. J. Math. 86 (1964), 1-16. MR 29 #4060. MR 0166787 (29:4060)
  • [8] W. S. Massey, Some higher order cohomology operations, Sympos. Internacional de Topologia Algebraica, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 145-154. MR 20 #4826. MR 0098366 (20:4826)
  • [9] J. P. May, Matric Massey products, J. Algebra 12 (1969), 533-568. MR 39 #289. MR 0238929 (39:289)
  • [10] R. M. F. Moss, Secondary compositions and the Adams spectral sequence, Math. Z. 115 (1970), 283-310. MR 42 #1123. MR 0266216 (42:1123)
  • [11] S. P. Novikov, The methods of algebraic topology from the point of view of cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 855-951. (Russian) MR 36 #4561. MR 0221509 (36:4561)
  • [12] G. J. Porter, Higher products, Trans. Amer. Math. Soc. 148 (1970), 315-345. MR 41 #1053. MR 0256397 (41:1053)
  • [13] N. Ray and R. M. Switzer, On $ {\text{SU}} \times {\text{SU}}$ bordism, Quart. J. Math. 21 (1970), 137-150. MR 0264679 (41:9270)
  • [14] D. M. Segal, On the symplectic cobordism ring, Comment. Math. Helv. 45 (1970), 159-169. MR 41 #7699. MR 0263094 (41:7699)
  • [15] E. H. Spanier, Higher order operations, Trans. Amer. Math. Soc. 109 (1963), 509-539. MR 28 #1622. MR 0158399 (28:1622)
  • [16] R. E. Stong, Notes on cobordism theory, Math. Notes, Princeton Univ. Press, Princeton, N. J.; Univ. of Tokyo Press, Tokyo, 1968. MR 40 #2108. MR 0248858 (40:2108)
  • [17] H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies, no. 49, Princeton Univ. Press, Princeton, N. J., 1962. MR 26 #777. MR 0143217 (26:777)
  • [18] H. Uehara and W. S. Massey, The Jacobi identity for Whitehead products, Algebraic Geometry and Topology, Princeton Univ. Press, Princeton, N. J., 1957, pp. 361-377. MR 19, 974. MR 0091473 (19:974g)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55G30

Retrieve articles in all journals with MSC: 55G30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0293623-1
Keywords: Cobordism, Massey product, manifolds, Pontrjagin-Thom map, Adams spectral sequence
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society