Math 662 Homework Assignment 4 (Optional)
Due Wednesday, December 4
1. Formulate a statement of a theorem on balancing Tor and prove it.
2. Let n be a positive odd integer. Find the cohomology of the dihedral
group of order 2n with coefficients in the integers.
3. Finish the following examples from class:
(a) Let G be a group of order 2 and k a field of characteristic 2.
Show that the group cohomology H^*(G,k) is isomorphic to the
polynomial ring k[x] under cup product, where deg(x)=1.
(b) Let k be a field of characteristic not 2. Show that the
cohomology H^*(k[x,y],k) is isomorphic to the exterior algebra
on a vector space of dimension 2, under cup product. Generalize
to a polynomial ring in n indeterminates.