Geometriae Dedicata

, Volume 148, Issue 1, pp 371–389 | Cite as

Controlling indeterminacy in Massey triple products

  • Laurence R. Taylor
Original Paper


A technique is discussed to control the indeterminacy in cohomology Massey triple products. It involves finding cohomology classes with certain properties. Poincaré duality spaces always have such classes if the coefficients are in a field.

A variety of non-vanishing and vanishing results for Massey triple products are proved using this technique. Here are two examples.
  1. Many authors have noticed that non-trivial triple products in a submanifold produce non-trivial triple products in the blowup along the submanifold.

  2. Given a map of closed, compact manifolds of the same dimension, \({f\colon M \to N}\) , then non-trivial triple products with field coefficients in N pull back to non-trivial triple products in M provided the degree of the map is non-zero in the field.



Massey triple product Poincaré duality 

Mathematics Subject Classifications (2000)

55S30 57P10 


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  1. 1.
    Donaldson S.K.: Symplectic submanifolds and almost-complex geometry. J. Diff. Geom. 44(4), 666–705 (1996) MR 1438190 (98h:53045)MATHMathSciNetGoogle Scholar
  2. 2.
    Ekedahl, T.: Two examples of smooth projective varieties with nonzero Massey products, Algebra, algebraic topology and their interactions (Stockholm, 1983). Lecture Notes in Mathematics, vol. 1183, pp. 128–132. Springer, Berlin (1986). MR 846443 (87f:14006)Google Scholar
  3. 3.
    Johnson, F.E.A., Rees, E.G.: The fundamental groups of algebraic varieties, Algebraic topology Poznań 1989. Lecture Notes in Mathematics, vol. 1474, pp. 75–82. Springer, Berlin, (1991). MR 1133893 (92j:14025)Google Scholar
  4. 4.
    Katz M.: Systolic inequalities and Massey products in simply-connected manifolds. Israel J. Math. 164, 381–395 (2008) MR 2391156MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kraines D.: Massey higher products. Trans. Am. Math. Soc. 124, 431–449 (1966) MR 0202136 (34 #2010)MATHMathSciNetGoogle Scholar
  6. 6.
    Lang, S.: Algebra. 2, Addison-Wesley Publishing Company Advanced Book Program, Reading, (1984). MR 783636 (86j:00003)Google Scholar
  7. 7.
    Massey W.S.: Higher order linking numbers. J. Knot Theory Ramifications. 7(3), 393–414 (1998) MR 1625365 (99e:57016)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    May J.P.: Matric Massey products. J. Algebra. 12, 533–568 (1969) MR 0238929 (39 #289)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    McDuff D.: Examples of simply-connected symplectic non-Kählerian manifolds. J. Diff. Geom. 20(1), 267–277 (1984) MR 772133 (86c:57036)MATHMathSciNetGoogle Scholar
  10. 10.
    Milgram R.J.: Steenrod squares and higher Massey products. Bol. Soc. Mat. Mexicana 13(2), 32–57 (1968) MR 0263074 (41 #7679)MATHMathSciNetGoogle Scholar
  11. 11.
    Rudyak Y., Tralle A.: On Thom spaces, Massey products, and nonformal symplectic manifolds. Internat. Math. Res. Notices 10, 495–513 (2000) MR 1759504 (2001h:53128)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Rusin D.J.: The cohomology of the groups of order 32. Math. Comp. 53(187), 359–385 (1989) MR 968153 (89k:20078)MATHMathSciNetGoogle Scholar
  13. 13.
    Uehara, H., Massey, W.S.: The Jacobi identity for Whitehead products, Algebraic geometry and topology. In: A symposium in honor of S. Lefschetz, pp. 361–377. Princeton University Press, Princeton, (1957). MR 0091473 (19,974g)Google Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsNotre Dame UniversityNotre DameUSA

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