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.

Definition: A CW spectrum is a collection of CW spaces $\{X_n\}$ for integers $n$ together with a collection of maps of CW complexes $\Sigma X_n \rightarrow X_{n+1}$ that are inclusions of subcomplexes.  A spectrum $X'$ is a subspectrum of $X$ if for each $n$ we have that $X_n' \subseteq X_n$, and $\Sigma X'_n \rightarrow X'_{n+1}$ is the restriction of the map $\Sigma X_n \rightarrow X_{n+1}$ to $\Sigma X'_n$.

Notice that a non-basepoint $k$-cell of $X_n$ suspends to a $(k+1)$-cell of $X_{n+1}$.  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 $f: X \rightarrow Y$ as a collection of cellular maps $f_n: X_n \rightarrow Y_n$ which fit into commutative squares with the suspension maps and the maps $\Sigma f_{n-1}: \Sigma X_{n-1} \rightarrow \Sigma Y_{n-1}$.

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 $X'$ of $X$ to be cofinal in $X$ if for every $n$, every cell $e_{\alpha}^i$ of $X_n$ is such that there is some large enough $k$ such that $\Sigma^k e_{\alpha}^i$ is a cell in $X'_{n+k}$.  In other words, every cell of a space in $X$ eventually suspends to a cell of a space in $X'$.

Now we define a map of CW spectra $f: X \rightarrow Y$ to be an equivalence class of strict maps $X' \rightarrow Y$ for cofinal subspectra $X'$ of $X$, where we regard two maps $f_1: X' \rightarrow Y$ and $f_2: X'' \rightarrow Y$ as equivalent if there is a subspectrum $X'''$ cofinal in both $X'$ and $X''$ such that $f_1$ and $f_2$ agree on $X'''$.

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

We define a homotopy of maps of CW spectra, $f_1: X \rightarrow Y$ and $f_2: X \rightarrow Y$, to be a map of spectra $F: X \times I \rightarrow Y$ which is the map $f_i$ on $X \times \{i\}$, $i=0, 1$.  Here we regard $X \times I$ as a spectrum with $(X \times I)_n=X_n \times I$ 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 $X_n \times I$ is just the reduced product of the reduced suspension of $X_n$ with $I$.  We denote the homotopy classes of maps as $[X, Y]$.

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 $[X, Y] \rightarrow [\Sigma X, \Sigma Y]$ is an isomorphism.

Proof: To see that the above map is surjective, let $f: \Sigma X \rightarrow \Sigma Y$ be a map of CW spectra.  If $f$ is a strict map on a cofinal subspectrum $Z'$ of $\Sigma X$, we set $X'_n=Z_{n-1}$.  Then we have that the spectrum $X'$ is cofinal in $X$, the spectrum $\Sigma X'$ is cofinal in $\Sigma X$, and the map $f$ is strict on $\Sigma X'$.  In other words, we may assume that $f: \Sigma X \rightarrow \Sigma Y$ is strict in the first place.

Now we rewrite $\Sigma X=X \wedge S^1$ where $(X \wedge S^1)_n=X_n \wedge S^1$ and the map from $\Sigma(X_n \wedge S^1)$ to $X_{n+1} \wedge S^1$ is $\Sigma \wedge 1$.  So we have $f: X \wedge S^1 \rightarrow Y \wedge S^1$.  We may replace $f_n$ by its restriction

$\Sigma f_{n-1}: \Sigma(X_{n-1} \wedge S^1) \rightarrow \Sigma(Y_{n-1} \wedge S^1)$

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 $S^1$.  Thus, each $f_n$ will be replaced by a map $h_{n} \wedge 1$.  Then this map $h$ will be sent to $f$ via the homomorphism above.

So how can we homotope $\Sigma f_{n-1}$ to a map which is independent of the coordinate in $S^1$?  Well, we write $\Sigma(X_{n-1} \wedge S^1)= X_{n-1} \wedge \Sigma S^1$.  Now the map may depend on the coordinate in $S^1$, identified with the equator in $\Sigma S^1$.  We homotope the sphere $\Sigma S^1=S^2$ by rotating it by 90 degrees.  Now the new map, also called $f_n$, on this sphere is independent of the new equator, and we identify $X_{n-1} \wedge \Sigma S^1$ with $\Sigma(X_{n-1} \wedge S^1)=\Sigma X_{n-1} \wedge S^1$.  This new map is independent of both $\Sigma$ and $S^1$ coordinates (since $S^1$ is now the new equator inside $\Sigma S^1$, so it is of the form

$h_n \wedge 1: X_{n-1} \wedge \Sigma S^1=\Sigma(X_{n-1} \wedge S^1) \rightarrow \Sigma(Y_{n-1} \wedge S^1)$

The proof of injectivity of the above homomorphism is similar.  Let $f$ and $g$ be maps $X \rightarrow Y$ such that $\Sigma f$ and $\Sigma g$ are homotopic.  That is, there is a map of CW spectra

$F: \Sigma X \times I \rightarrow \Sigma Y$

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

But since the homotopy $F$ above is independent of the $\Sigma$ coordinate (that is, it sends each horizontal cross section of the suspension of $X_n$ into the horizontal cross section of $\Sigma Y_n$ with the same $\Sigma$ coordinate), it maps the copy of $X_n \times I$ in the middle of all of the suspensions into the copy of $Y_n$ in the middle of $\Sigma Y$.  This is a map of CW spectra restricts to $f$ and $g$ at the endpoints. $\Box$

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.

From → Uncategorized

If underlying spaces exist 😉

Nice post brah.

• Thanks dawg. You should post something something.

It might be helpful to mention that, given maps $\Sigma X_{n}\to X_{n+1}$, adjointness of $\Sigma$ and $\Omega$ implies existence of maps $X_{n}\to\Omega X_{n+1}$ where $\Omega X{n+1}$ is the loop space of $X_{n+1}$.
Erm…. that loop space should be $\Omega X_{n+1}$. Looks like I forgot an underscore.