I want to prove the following: given the (already proven) fact that the we have a bijection between (continuous maps) $f:X\rightarrow Y^K$ and $g:X\wedge K\rightarrow Y$ for pointed spaces $X$,$Y$ and $K$ with $K$ locally compact Hausdorff given by $g([x,k])=f(x)(k)$, then I want to prove that there is a bijection of homotopy classes $[X,Y^K]\cong[X\wedge K,Y]$. The hint i get is to use the cylinder construction (thus $X\mapsto X\times I$), that this is natural and that there are two natural inclusions $W\rightarrow W\times I$, namely on the bottom and on the top. Someone an idea to proof this?
I started with this suppose $f\cong h$ with $f,h:X\rightarrow Y^K$ (in the sense of homotopy), then there exists $H:X\times I\rightarrow Y^K$ with the well known properties. I want to construct now $H':(X\wedge K)\times I\rightarrow Y$, but i don't know how to do this. Someone an idea?
Thank you very much.
Consider the inclusion $\{0,1\}\hookrightarrow [0,1]=:I$ and its pointed version $\{0,1\}^+\hookrightarrow I^+$. You have $\text{C}_{\ast}(\{0,1\}^+\wedge A,B)\cong\text{C}_{\ast}(\{0\}^+\wedge A,B)\times\text{C}_\ast(\{1\}^+\wedge A,B)\cong\text{C}_\ast(A,B)\times\text{C}_\ast(A,B)$. Under this bijection, a pair $(f,g)$ of continuous, pointed maps $f,g: A\to B$ extends to a map $\text{C}_\ast(I^+\wedge A,B)$ if and only if $f$ and $g$ are pointed homotopic. Now, apply this twice, firstly to $A=X$ and $B=Y^K$, and secondly to $A=X\wedge K$ and $B = Y$. Using naturality of the bijection you already constructed, this gives you the statement about homotopy classes.