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?



What is a cup-product in group cohomology, and how does it relate to other branches of mathematics?
You can identify group cohomology with $Ext\_{kG}(k,k)$ where $k$ is your base ring, and $kG $ is the group algebra. Here, the cup product given by Yoneda product (which there's some material on [here](https://en.wikipedia.org/wiki/Ext_functor#Ring_structure_and_module_structure_on_specific_Exts)). You should think Ext as being a generalization of Hom (it measures maps that should be there and aren't, and higher order things along the same lines), and this product as a generalization of composition.


Almost anything anyone ever calls cohomology is Ext of something with something, and products come up very naturally this way. Cohomology of manifolds is Ext of the sheaf of locally constant k valued functions with itself, and the cup product again is just Yoneda product (all the explicit models for cohomology (simplicial, De Rham, Cech) are all just different resolutions of the this sheaf).


