Homology pp 220-248 | Cite as


  • Saunders Mac Lane
Part of the Classics in Mathematics book series (volume 114)


Throughout the study of products there is an interplay between “external” and “internal” products. This relation may be illustrated in the case of homology products. If X R and R Y are chain complexes of R-modules the external homology product os the homomorphism of abelian groups
$$p:{H_k}(X){ \otimes _R}{H_m}(Y) \to {H_{k + m}}(X{ \otimes _R}Y),$$
defined on cycles u of X and v of Y by
$$p({\text{cls }}u \otimes {\text{cls }}\upsilon ){\text{ = cls}}(u \otimes \upsilon ).$$


Exact Sequence Hopf Algebra Short Exact Sequence Simplicial Module Simplicial Object 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Saunders Mac Lane
    • 1
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

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