I just read Greenlees’ paper, Spectra for Commutative Algebraists. Spectra have long been one of my ‘favorite things in a subject far from my own’, though the above paper mentions techniques designed at removing that qualification.
Spectra (singular: spectrum) are a weakening of the notion of a topological space. The motivation is that the Freudenthal suspention theorem states that by successively suspending a finite CW complex, eventually the behavior ‘stabilizes’ and no new information is introduced. Concretely, suspension induces a map from to , and this map is an isomorphism if the suspension functor has already been applied enough times. The idea of spectra is to describe things akin to spaces, but which capture the stable information, rather than the pre-stable information of the actual spaces.
This is accomplished by defining a new category (the category of spectra), whose objects are CW complexes, but a map between and is given by a homotopy equivalence class of maps between and for some (I am omitting oodles of important details). Thus, two spectra represented by CW complexes and are isomorphic if they become homotopy equivalent after enough suspensions.
Spectra are great because they are central for several interesting results. First, they are manifestations of the stable homotopy groups of the CW complexes they represent. Second, the Brown representability theorem (one of the top 5 under-appreciated theorems) says that any cohomology theory on CW complexes satisfying the Eilenberg-Steenrod axioms is given by , for some spectrum .
It has always bothered me that I don’t have a better intuitive grasp of what passing to a spectrum ‘mods out’ by. Are there some simple moves (maybe akin to Reidermeister moves), which change the CW complex but leave the stable homotopy structure invariant? Maybe some surgeries? A simpler but equivalent question, can anyone think of two spaces which are not homotopy equivalent but whose suspensions are?