TITLE: What is a cup-product in group cohomology, and how does it relate to other branches of mathematics?
QUESTION [34 upvotes]: I have a few elementary questions about cup-products.
Can one develop them in an axiomatic approach as in group cohomology itself, and give an existence and uniqueness theorem that includes an explicitly computable map on cochains?  Second, how do they relate to cup-products in algebraic topology?  In general, are there connections between cup-products and other mathematical constructions that may provide more intuition into them?

REPLY [3 votes]: You should definitely take a look at Lang's "Topics in Cohomology of Groups", chapter 4. There a general notion of cup-products on delta-functors is introduced. This may look like a lot of abstract nonsense to most people, but I like it :)