Introduction to Part IV
We have a number of means at our disposal for systematically calculating the values of homology and cohomology modules. No such general devices are available for cup products. In practice two broad lines of attack are perceptible. On the one hand, special structure on the space in question may place restrictions on the possible cohomology algebras. For manifolds, the duality theorems impose strong restrictions. On the other hand, by working in an abstract setting, one can construct auxiliary products and determine many relations among them. These in turn can often be used to determine the cohomology algebra. This part develops some basic aspects of the formal procedure.