Abstract
In this chapter we introduce the theory of products in homology and cohomology. Following the Künneth formula for free chain complexes, we state and prove the acyclic model theorem. This is applied to establish the Eilenberg-Zilber theorem and the resulting external products in homology and cohomology. When the coefficient group is a ring R, it is shown that the cohomology external product may be refined to the cup product, giving the cohomology group the structure of an R-algebra. This structure is computed for the torus by introducing the Alexander-Whitney diagonal approximation. Also, a cup product definition of the Hopf invariant is given. Finally, the cap product between homology and cohomology is defined in anticipation of Chapter 6.
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© 1994 Springer Science+Business Media New York
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Vick, J.W. (1994). Products. In: Homology Theory. Graduate Texts in Mathematics, vol 145. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0881-5_5
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DOI: https://doi.org/10.1007/978-1-4612-0881-5_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6933-5
Online ISBN: 978-1-4612-0881-5
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