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?

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September 8, 2007 at 6:42 am |

Spectra aren’t best thought of as a “weakening of the notion of topological space”. They’re really a weakening of the notion of

abelian topological groupThis weakening has two features, and it’s best to consider them separately.The first and most important weakening is that in a spectrum, all the usual laws of an abelian group hold

up to homotopy, with these homotopies satisfying all the right coherence laws (like the pentagon identity for associativity and the hexagon identity for commutativity)up to homotopy, ad infinitum.If we only weaken the concept of abelian topological group in this way, we arrive at the notion of

infinite loop space, or, what’s almost but not quite the same,connective spectrumorE∞ space. The slight differences between these concepts are best tackled by reading J. F. Adams’ wonderful bookInfinite Loop Spaces. If you’re too cheap to buy that, try the relevant issues ofThis Week’s Finds, mainly “week149”, “week150″ and week199”.The second weakening allows for a spectrum to have nontrivial nth homotopy groups for

negativevalues of n, not just n ≥ 0. This is like passing from N-graded chain complexes to Z-graded chain complexes. (In fact, an N-graded chain complex is just another way of looking at an abelian topological group, so the analogy is quite precise.)The second weakening means that a spectrum is not just a space with extra structure anymore.

Spectra are designed to be gadgets that let you understand generalized cohomology theories in a conceptual way (see “week149”), and to get generalized cohomology theories where spaces have nontrivial negative cohomology groups, we need this second weakening.

In short, I think you’re asking very tough questions, like “when do two spaces give the same suspension spectrum”, instead of the very nice questions that led people to invent spectra in the first place. I can’t recommend Adams’ book too highly as a source of good intuition about spectra.

September 8, 2007 at 10:43 pm |

Hey,

Spectra can also be thought of as the “linearization” or “abelianization”

of spaces (of course this agrees with what John explained above, thinking of spectra as cohomology theories).

Namely we want to take the category of spaces and make it into an additive category – or more precisely something whose homotopy category is additive. (It’s useful for people brought up in algebraic geometry,

like me, to remember that the homotopy category of spectra is a

triangulated category with a t-structure whose heart is abelian groups —

in other words spectra differ from complexes of abelian groups in that their

higher Ext groups are not generated by Ext^1s, but involve things

like the stable homotopy groups of spheres —- rationally spectra are the same as graded vector spaces however.)

The answer to the problem of abelianization is that

we need to localize by (ie invert) the functor of suspension,

resulting in spectra – it’s useful to think of spectra as a localization

of spaces, much as we think of localizing a ring (and to remember that the idea of localization is to make things easier, ie throw out some

complicated information).

There’s a precise notion of linearization

in absurdly general contexts – in fact there’s a full theory of calculus

(due to Goodwillie) where linearization is of course the first derivative –

and spectra arise (as far as I understand) by linearizing the category

of spaces in this sense. Another example of linearization in this sense

are the linearization of the category of rings/R to the derived category of R-modules (Quillen’s construction of the tangent complex).

September 9, 2007 at 2:32 am |

Thanks for the advice!

I am surprised that you both seemed to come out rather strongly against thinking of spectra as weakened spaces. Certainly the group structure is mysterious, but its only literally untrue if we allow spectra which are non-trivial in arbitrarily negative degrees, or possibly some spectra that keep getting more complicated in positive degrees. I was under the impression that tossing out such examples lost some desirable objects (the K-theory spectra, for instance), but otherwise was about as forgivable as working in the bounded derived category rather than the full derived category.

September 12, 2007 at 2:05 pm |

There is an exercise in Hatcher’s book Algebraic Topology on page 389 which ask one to show that that suspension of any acyclic CW complex is contractible. This gives a wealth of examples If you look at wikipedia, you’ll see that you can make such a space for every perfect group. This also leads to some of the sorts of “Reidermeister moves” you might be looking for. Every time you find a piece of your CW complex which looks like an acyclic piece, you can kill it.

However there are lots and lots of other relations. I think the following gives an easy example of two spaces which become homotopic after suspending: If you take the torus and suspend it you get a space which is (homotopic to) the wedge of a 2-sphere, another 2-sphere, and a 3-sphere. Hatcher’s book also explains this. I believe that if you suspend the wedge of a circle, another circle and a 2-sphere you get a homotopy equivalent space. You can check for yourself.

September 12, 2007 at 6:52 pm |

Great! Thats just the kind of example I was looking for. Its probably an impossible task to get an exhaustive set of tools for moving between equivalent spectra, but I am happy to know that there are examples that are that concrete and explainable.

October 1, 2007 at 5:44 am |

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