Why do $S^1 \wedge - $ and $Maps(S^1,-)$ form a Quillen adjunction?

Question

Consider the category of pointed topological spaces $Top^{*/}$ with the standard Quillen model structure. I understand that $S^1 \wedge - $ and $Maps(S^1,-)$ are a pair of adjoint functors. How do I now prove that they are actually a Quillen adjunction ?

Answer 1

You need to show that the left adjoint ($S^1\wedge-$) preserves cofibrations and trivial cofibrations, and that the right adjoint $Map_*(S^1,-)$ preserves fibrations, trivial fibrations and limits.

We'll work in the convenient category of based, compactly generated spaces to make the question meaningful. We'll work on the left adjoint first, and then most of the required properties for the right adjoint will follow by immediately. Note that our category is cofibrantly generated, with generating cofibrations the set of maps $S^{n-1}_+\hookrightarrow D^n_+$, and generating acyclic cofibrations $D^n_+\hookrightarrow (D^n\times I)_+$, thus to demonstrate that $S^1\wedge-$ has the required properties it will suffice to demonstrate it preserves these cofibrations and trivial cofibrations.

We easily see that $S^1\wedge S^n_+\cong S^{n+1}\vee S^1\hookrightarrow S^1\wedge D^n_+\cong D^{n+1}\vee S^1$ is a cofibration for each $n\geq 0$, and from this it follows that $S^1\wedge D^n_+\hookrightarrow S^1\wedge(D^n\times I)_+\cong S^1\wedge (D^n_+\wedge I_+)\cong (S^1\wedge D^n_+)\wedge I_+$ is a cofibration and a homotopy equivlence for each $n\geq 0$. In short $S^1\wedge-$ is left Quillen.

Now to check that $Map_*(S^1,-)$ is right Quillen we show that it preseves fibrations by using adjunction to show that it $Map(S^1,p)$ has the right lifting propery against all acyclic cofibrations for any fibration $p:E\rightarrow B$. Since $Map(S^1,-)$ is obviously a homotopy functor it will preseve acyclic fibrations by the same argument. To check that it preseves limits, simply note that it is representable.

Thus we have a Quillen adjunction.