User:Alodyne/Stable homotopy theory

In mathematics, stable homotopy theory is a branch of algebraic topology. Stable homotopy theory differs from classical homotopy theory in that the phenomena under consideration are in some way independent of dimension. More precisely, stable homotopy theory isolates the phenomena of homotopy theory that are unchanged after sufficiently many applications of suspension. The theory has been successful in that many difficult problems have more tractable stable analogues, and that one may often use stable methods to treat unstable cases. The theory was pioneered during the period beginning at the end of World War II and ending in the early 1970s, by Adams, Atiyah, Boardman, Cartan, Eilenberg, Grothendieck, Milnor, Moore, Pontryagin, Postnikov, Quillen, Serre, Steenrod, and others. More recently, the machinery of stable homotopy theory has been used to great effect in algebraic geometry, modular representation theory, and noncommutative geometry. In particular, Vladimir Voevodsky's solution of the famous Milnor conjecture, for which he was awarded the Fields Medal in 2002, is a successful installation of the framework of modern stable homotopy theory in an algebro-geometric context. Notable theorems of modern stable homotopy theory include the nilpotence theorem and its consequence the periodicity theorem, both due to E.S. Devinatz, M.J. Hopkins, and J. Smith.

The most famous important problem in stable homotopy theory is the computation of the stable homotopy groups of spheres. This problem is also an example of how topological homotopy theories are more complicated than algebraic ones: in most algebraic homotopy theories, the analogous computation is trivial.

Introduction
The Freudenthal suspension theorem leads to several important results that depend heavily on connectivity hypotheses. Its corollaries, which are numerous and significant, thus also depend on such hypotheses. The rough region in the dimension-connectivity plane in which these hypotheses are satisfied is informally referred to as the stable range, which is the origin of the phrase stable homotopy theory. In this range, cofibrations and fibrations coincide; other nice properties... Any map f: X &rarr; Y becomes stable after enough suspensions. Each application of suspension increases both dimension and connectivity by one, so it is clear that all maps of spaces are thus "eventually stable".

In the modern treatment of stable homotopy, the category (respectively, homotopy category) of topological spaces is typically replaced by the category (respectively, homotopy category) of spectra. A spectrum is to a space what a chain complex is to an abelian group. Spectra act like spaces in many ways, or rather like very nice spaces. In particular, the suspension functor and its left adjoint the loop space functor induces a pair of adjoint functors on the category of spectra, which is an autoequivalence on the homotopy category.