Basic Algebraic Topology

Chapter 6 Universal Coefficient Theorem for Homology

In this short chapter, we introduce homology groups with coefficients in an arbitrary module G over a given ring R and establish some algebraic relations between these groups and the homology groups with coefficients in R. On the way, we introduce the powerful and almost only known method to establish natural equivalences of various functors taking values in the category of chain complexes, viz., the method of acyclic models. As an easy consequence we get a proof of Theorem 4.3.3.

As a natural generalization, we then obtain the so-called Künneth formula for the homology of the tensor product of two chain complexes. With the help of Eilenberg–Zilber map this is then converted to a formula relating ...