ABSTRACT
This chapter provides an algebraic device that will be used for studying the homology of a product space. The direct product of sequences of the first type gives a sequence that is naturally isomorphic with the second sequence in its three right-hand terms. A question arises: can one also introduce Ext and torsion product for modules? The answers to this question constitute, basically, the subject matter of homological algebra. The product functors form part of a general subject called Homological Algebra. There is a similar sequence for cohomology that holds if the homology is finitely generated. The chapter shows how the homology groups of a space determine the cohomology groups. In differential geometry, it is common to deal with compact manifolds and to use the field of reals as coefficients. Because the cohomology and homology vector spaces are dual in this case, differential geometers sometimes treat homology and cohomology as if they were the same object.
