Chapter 6 - Modern Foundations for Stable Homotopy Theory

Abstract

The stable homotopy theory began around 1937 with the Freudenthal suspension theorem. In simplest terms, it states that, if q is small relative to n, then πn+q(Sn) is independent of n. Stable phenomena had of course appeared earlier, at least implicitly. Reduced homology and cohomology are examples of functors that are invariant under suspension without limitation on dimension. The stable homotopy theory emerged as a distinct branch of algebraic topology with Adams' introduction of his eponymous spectral sequence and his spectacular conceptual use of the notion of stable phenomena in his solution to the Hopf invariant one problem. Its centrality was reinforced by two related developments that occurred at very nearly the same time, in the late 1950's. The stable homotopy theory demands a category in which to work. One may set up the ordinary Adams spectral sequence ad hoc, as Adams did, but it would be ugly at best to set up the Adams spectral sequence based on a generalized homology theory that way.