Products in Homology and Cohomology

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


The most important product is undoubtedly the so-calledcup product: It assigns to any elements u ∈H p (X; G1) and v ∈ Hq(X ; G2) an elementu ∪ v ∈ H p + q (X ;G1⊗ G2). This product is bilinear (or distributive), and is natural with respect to homomorphisms induced by continuous maps. It is an additional element of structure on the cohomology groups that often allows one to distinguish between spaces of different homotopy types, even though they have isomorphic homology and cohomology groups. This additional structure also imposes restrictions on the possible homomorphisms which can be induced by continuous maps.


Cross Product Chain Complex Cohomology Group Finite Rank Free Abelian Group 
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Bibliography for Chapter VIII

  1. [1]
    A. Dold,Lectures on Algebraic Topology, Springer-Verlag, New York, 1972.MATHGoogle Scholar
  2. [2]
    S. Eilenberg and N. Steenrod,Foundations of Algebraic Topology, Princeton University Press, Princeton, 1952.MATHGoogle 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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