Theorems of periodicity and approximation in homological algebra

Abstract

Let us begin by contrasting the spectral sequence I have developed with the classical method of killing homotopy groups, as applied to the calculation of stable homotopy groups. Both depend on a knowledge of the stable Eilenberg-NacLane groups H^n+q (π, n; G) (n > q) for some π and G . Neither of them is an algorithm. By an algorithm I would mean a procedure that comes provided with a guarantee that you can always compute any required group by doing a finite amount of work following the instructions blindly. In the case of the method of killing homotopy groups, you have no idea how far you can get before you run up against some ambiguity and don’t know how to settle it. In the case of the spectral sequence, the situation is clearer: the groups Ext ^{s, t}_A (H^✲(Y), H^✲(X)) are recursively computable up to any given dimension; what is left to one’s intelligence is finding the differentials in the spectral sequence, and the group extensions at the end of it.