Show that for Abelian groups $G$ and $H, \bigl[K(G, n), K(H, n)\bigr] \cong \operatorname{Hom}(G, H).$

Question

Show that for Abelian groups $G$ and $H, \bigl[K(G,n), K(H,n))\bigr] \cong \operatorname{Hom}(G, H).$

I was given a hint to use the following:

But I have many stupid questions:

1-First, how I am going to use these all things in the proof?

2-Second, How I am going to prove the theorem given in the hint?

3-How am I going to prove the problem given?

4-Is there solutions to some of my problems here in this site?

we are using the book of "Modern Classical Homotopy Theory" by Jeffery Strom.

Answer 1

I'd prefer to give you a few more hints, rather than a complete solution. If you are still stuck with anything, let me know in the comments where, and I'll fill in some more details. I'll assume that $n>1$ below to avoid some techincal details (I'm not sure the statement is correct without this assumption).

$(a)$ Assume that $X$ is a based CW complex. For each $k\geq 0$ the $(k+1)$-skeleton $X_{k+1}$ is obtained from the $k$-skeleton $X_k$ by attaching $(k+1)$-cells. That is, there is a cofibration sequence of the form

$$\bigvee S^{k}\rightarrow X_{k}\rightarrow X_{k+1}\rightarrow \bigvee S^{k+1}\rightarrow \dots$$

Note that $\pi_kK(G,n)=0$ for $k\neq n$. Study the resulting Puppe sequence (Corollary 8.4) and use induction.

$(b)$ If $X$ is $(n-1)$-connected then you can assume that $X_n\simeq\bigvee S^n$ (see $\S$ 16.1).

$(c)$ The inclusion $X_n\hookrightarrow X$ is an $n$-equivalence for any complex $X$ (this is again $\S$ 16.1).

$(Aside)$ It might pay to observe a few facts at this point: i) If $X=S^n$ then $\phi$ is already an isomorphism (this is immediate from the definition of a $K(G,n)$). ii) $\pi_n(\bigvee S^n)\cong \bigoplus \pi_nS^n$. iii) Thus if $X=\bigvee S^n$, then $\phi$ is an isomorphism. iv) In particular, $\phi:[X_n,K(G,n)]\rightarrow Hom(\pi_nX_n,G)$ is bijective for any $(n-1)$-connected complex $X$.

$(e)$ The homomorphism $h$ determines a homomorphism $\pi_n(X_n)\rightarrow\pi_nX=H\xrightarrow{h} G$, where the first map is a surjection. We know from earlier that $X_n\simeq \bigvee S^n$ and that $[X_n,K(G,n)]\cong Hom(\pi_nX_nG)$.

$(f)$ We have the cofibration sequence

$$\bigvee S^n\xrightarrow\alpha\bigvee S^n\xrightarrow{i} X$$

and a map $\beta:\bigvee S^n\rightarrow K(G,n)$ will extend over $X$ if and only if the composition $\beta\alpha$ is null-homotopic. If $\beta$ represents $h i_*$, then $\phi(\beta\alpha)=(\beta\circ\alpha)_*=\beta_*\alpha_*=hi_*\alpha_*=h(i\alpha)_*=h0=0$. We know from earlier that $\phi(\beta\alpha)=0$ is equivalent to $\beta\alpha\simeq\ast$.

$(d)$ If $X$ is a CW complex then $X\times I$ is a CW complex. If $\dim X=n+1$, then $\dim(X\times I)=n+2$ and $(X\times \partial I)\cup (X_n\times I)$ is an $(n+1)$-dimensional subcomplex. Assume $X$ is $(n-1)$-connected and apply Theorem 16.27.