Is Eilenberg-Maclane $\wedge$ Moore space the spectrum of the cohomology theory $H^*(\ ,G)$?

Question

In the web page http://www.encyclopediaofmath.org/index.php/Moore_space it can be found the following statement:

If $K(\mathbb Z,n)$ is the Eilenberg–MacLane space of the group of integers $\mathbb Z$ and $M_k(G)$ is the Moore space with $\tilde{H}_k(M_k(G))=G$, then

$$lim_{N\rightarrow\infty}[\Sigma^{N+k}X,K(\mathbb Z, N+n)\bigwedge M_k(G)]\cong H^n(X,G)$$,

that is, $\{K(\mathbb Z,n)\wedge M_k(G)\}$ is the spectrum of the cohomology theory $H^*(\ ,G)$.

The reference of the webpage is the articule of Moore ´´On homotopy groups of spaces with single non-vanishing homotopy group´´ but I don´t find anything like that on the article.

It would be very helpful if you could give a reference where I can see the explanation or you can explain me that.

Answer 1

Let $H\mathbb{Z}$ denote the spectrum for cohomology with coefficients in $\mathbb{Z}$, so your spectrum is $H\mathbb{Z}\wedge M_k(G)$. Let $$0\to\bigoplus_I\mathbb{Z}\stackrel{f}{\to}\bigoplus_J\mathbb{Z}\to G\to 0$$ be a presentation of $G$. Then we can explicitly construct a Moore space $M_k(G)$ as the cofiber of a map $\bigvee_I S^k\to \bigvee_J S^k$ whose induced map on homology is $f$. In spectra, we then have a cofiber sequence $$\bigvee_I \Sigma^k H\mathbb{Z}\to\bigvee_J\Sigma^k H\mathbb{Z}\to H\mathbb{Z}\wedge M_k(G).$$

For any pointed finite CW-complex $X$, we then have a long exact sequence in cohomology of the form $$\dots\to\bigoplus_IH^{n+k}(X,\mathbb{Z})\to \bigoplus_J H^{n+k}(X,\mathbb{Z})\to H\mathbb{Z}\wedge M_k(G)^n(X)\to\dots$$

Now consider $X=S^0$. The long exact sequence above computes that $H\mathbb{Z}\wedge M_k(G)^{-k}(S^0)=G$ and $H\mathbb{Z}\wedge M_k(G)^n(S^0)=0$ for $n\neq -k$. Thus up to a degree shift, the cohomology theory $H\mathbb{Z}\wedge M_k(G)$ satisfies the Eilenberg-Steenrod dimension axiom, so it must coincide with $H^{*+k}(X,G)$.

Answer 2

In case you are curious, here is a more concise but higher level argument.

The homotopy groups of the spectrum $H\mathbb Z \wedge M_k(G)$ agree with the homology groups of $M_k(G)$, which is just a $G$ in degree $k$. So the spectrum $H\mathbb Z \wedge M_k(G)$ is equivalent a shift of $HG$.

These are all standard facts in stable homotopy theory; Eric's answer above goes into more detail as to why they are true in this case.