Finiteness Conditions

Abstract

Recall that the definition of H_*(G, M) and H*(G, M) allows us to choose an arbitrary projective resolution P = (P_i)_i≥0 of ℤ over ℤG. Similarly, if we wish to take the topological point of view, then we can compute H_*(G, M) and H*(G, M) in terms of an arbitrary K(G, 1)-complex Y. Since we have this freedom of choice, it is reasonable to try to choose P (or Y) to be as “small” as possible, and this leads to various finiteness conditions on G.