Cup Products in Projective Spaces and Applications of Cup Products

  • William S. Massey
Part of the Graduate Texts in Mathematics book series (GTM, volume 70)


In this chapter we will determine cup products in the cohomology of the real, complex, and quaternionic projective spaces. The cup products (mod 2) in real projective spaces will be used to prove the famous Borsuk—Ulam theorem. Then we will introduce the mapping cone of a continuous map, and use it to define the Hopf invariant of a map f : S 2n-1 → S n . The proof of existence of maps of Hopf invariant 1 will depend on our determination of cup products in the complex and quaternionic projective plane.


Projective Space Commutative Diagram Quotient Space Mapping Cone Mapping Cylinder 
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Bibliography for Chapter X

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    J. F. Adams, On the nonexistence of elements of Hopf invariant one, Ann. Math., 72 (1960), 20–104.MATHCrossRefGoogle Scholar
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    J. Adem, The iteration of Steenrod squares in algebraic topology, Proc. Nat. Acad. Sci., 38 (1952), 720–726.MathSciNetMATHCrossRefGoogle Scholar
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    N. Bourbaki, Topologie Générale, Hermann et Cie., Paris, 1947, Chapters VI and VIII.MATHGoogle Scholar
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    H. Freudenthal, Oktaven, Ausnahme-gruppen, und Oktavengeometrie (mimeographed), Utrecht, 1951, revised ed., 1960.Google Scholar
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    G. W. Whitehead, On the Freudenthal theorems, Ann. of Math., 57 (1953), 209–228.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1980

Authors and Affiliations

  • William S. Massey
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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