# Spectra, part I

I have been trying to work out the details of what spectra are and the various constructions one can make with them. Today I’m going to define spectra and say a little bit about why they are useful. In the future, I’d like to continue this series of posts to talk about Brown representability, Eilenberg-MacLane spectra, the Adams spectral sequence, and maybe some other topics. Today’s notes are mostly from chapter 2 of Hatcher’s book on spectral sequences.

Let’s start with a few definitions.

**Definition: **A **CW spectrum **is a collection of CW spaces for integers together with a collection of maps of CW complexes that are inclusions of subcomplexes. A spectrum is a **subspectrum** of if for each we have that , and is the restriction of the map to .

Notice that a non-basepoint -cell of suspends to a -cell of . Now there is sort of a subtle question of how to define maps between CW spectra. Our first guess would be to define what we’ll call a **strict map** as a collection of cellular maps which fit into commutative squares with the suspension maps and the maps .

But it turns out that a nicer and weaker condition will let us do the things that we want (for example, maps of CW spectra should induce maps on the homology, cohomology, and homotopy groups of spectra, when we define them).

First, we define a subspectrum of to be **cofinal** in if for every , every cell of is such that there is some large enough such that is a cell in . In other words, every cell of a space in eventually suspends to a cell of a space in .

Now we define a **map of CW spectra** to be an equivalence class of strict maps for cofinal subspectra of , where we regard two maps and as equivalent if there is a subspectrum cofinal in both and such that and agree on .

It is a useful exercise to check that composing two maps of CW spectra and gives us a map of CW spectra. The idea is to choose a cofinal subspectrum *of the cofinal subspectrum* on which is defined, such that has the property that each of the cells in each of its complexes maps into , where is the cofinal subspectrum on which is defined. Then we may compose the restriction of to with , and it just remains to check that is cofinal in (it is because it is cofinal in ).

We define a **homotopy of maps of CW spectra, ** and , to be a map of spectra which is the map on , . Here we regard as a spectrum with and we mean the reduced product, with the interval above the basepoint collapsed to a point so that we have that the reduced suspension of is just the reduced product of the reduced suspension of with . We denote the homotopy classes of maps as .

Now we come to a proposition which explains what the homotopy category of spectra is good for and how it is different from the category of CW spaces. This proposition says that the suspension functor is invertible.

**Proposition: **The map is an isomorphism.

**Proof:** To see that the above map is surjective, let be a map of CW spectra. If is a strict map on a cofinal subspectrum of , we set . Then we have that the spectrum is cofinal in , the spectrum is cofinal in , and the map is strict on . In other words, we may assume that is strict in the first place.

Now we rewrite where and the map from to is . So we have . We may replace by its restriction

As we noted above, this map is independent of the coordinate “in” $\latex \Sigma$. We want to homotope it to a map that is also independent of the coordinate in . Thus, each will be replaced by a map . Then this map will be sent to via the homomorphism above.

So how can we homotope to a map which is independent of the coordinate in ? Well, we write . Now the map may depend on the coordinate in , identified with the equator in . We homotope the sphere by rotating it by 90 degrees. Now the new map, also called , on this sphere is independent of the new equator, and we identify with . This new map is independent of both and coordinates (since is now the new equator inside , so it is of the form

The proof of injectivity of the above homomorphism is similar. Let and be maps such that and are homotopic. That is, there is a map of CW spectra

restricting to and on the two endpoints. As above, we may assume that is strict. We need to come up with a homotopy . We can find inside by choosing the copy of that lies in the middle of each suspension. This gives us a map $\latex X \times I \rightarrow \Sigma Y$.

But since the homotopy above is independent of the coordinate (that is, it sends each horizontal cross section of the suspension of into the horizontal cross section of with the same coordinate), it maps the copy of in the middle of all of the suspensions into the copy of in the middle of . This is a map of CW spectra restricts to and at the endpoints.

The invertibility of the suspension functor essentially says that the homotopy category of spectra is the place in which to study stable phenomena. As we will see later, the homology/cohomology/homotopy groups of spectra will be the stable homology/cohomology/homotopy groups of the underlying spaces.

If underlying spaces exist 😉

Nice post brah.

Thanks dawg. You should post something something.

Also, one note:

It might be helpful to mention that, given maps , adjointness of and implies existence of maps where is the loop space of .

Erm…. that loop space should be . Looks like I forgot an underscore.