Stable Algebraic Topology and Stable Topological Algebra

Algebraic topology is a young subject, and its foundations are not yet firmly in place. I shall give some history, examples, and modern developments in that part of the subject called stable algebraic topology, or stable homotopy theory. This is by far the most calculationally accessible part of algebraic topology, although it is also the least intuitively grounded in visualizable geometric objects. It has a great many applications to such other subjects as algebraic geometry and geometric topology. Time will not allow me to say as much as I would like about that. Rather I will emphasize some foundational issues that have been central to this part of algebraic topology since the early 1960’s, but that have only been satisfactorily resolved in the last few years. It was only in 1952, with Eilenberg and Steenrod’s book “Foundations of algebraic topology” [9], that the nature of ordinary homology and cohomology theories was reasonably well understood. Even then, the modern way of thinking about cohomology as represented by Eilenberg-Mac Lane spaces was nowhere mentioned. It may have been known by then, but it certainly was not known to be important. The subject changed drastically with a series of extraordinary advances in the 1950’s and early 1960’s. By around 1960, it had become apparent that algebraic topology divides naturally into two rather different major branches: unstable homotopy theory and stable homotopy theory. The former concerns space level invariants, such as the fundamental group, that are more or less invisible to homology and cohomology theories. The latter concerns invariants that are in a sense independent of dimension. More precisely, it concerns invariants that are stable under suspension, such as homology and cohomology groups. It had also became apparent that many interesting phenomena that a priori seemed to depend on a dimension could be translated into questions in stable algebraic topology. Three fundamental examples have set the tone for a great deal of modern algebraic topology. They occurred nearly simultaneously in the late 1950’s and early 1960’s. The order I will give is not chronological. First, Adams [1] proved that the only possible dimensions of a normed linear algebra over R are 1, 2, 4, or 8 by translating the problem into one in stable homotopy theory. More precisely, the problem translated into a problem in ordinary mod 2 cohomology theory that involved only the Steenrod cohomology operations Sq : H(X;Z2) −→ H(X;Z2) and not the cup product. The Steenrod operations are stable, in the sense that ΣSq = SqΣ