Here is the question:
Show that $\langle ?,?\rangle$ (defined below) is natural in both variables. That is suppose $f\colon X \rightarrow Y,\ u \in \tilde{H^{*}}(Y),\ \alpha \in \tilde{H_{*}}(X).$ Then we can form $$ \langle u, f_{*}(\alpha)\rangle \in \tilde{H}_{n-k}(Y)$$ and $$\langle f^{*}(u), \alpha\rangle \in \tilde{H}_{n-k}(X).$$
Show that $$f_{*}(\langle f^{*}(u), \alpha \rangle) = \langle u, f_{*}(\alpha) \rangle. $$
The question depends on the following paragraph of pairing Cohomology with Homology in "Modern Classical Homotopy Theory " by Jeffery Strom:
So, we are not speaking about cap product it is just pairing. Could anyone help me in solving this, please?
Best Answer
If $\alpha \in \tilde{H}_{n}(X;B)$ then we can represent it with a homotopy class of maps $\alpha\colon S^{n+t} \to X\wedge K(B,t)$ for $t$ sufficiently large relative to $n$ (see Strom page 506 for his definition of ordinary homology). Given $f\colon X \to Y$, in order to get a class in $\tilde{H}_n(Y;B)$ we can form the composition $(f\wedge id_{K(B,t)})\circ \alpha\colon S^{n+t} \to Y \wedge K(B,t)$; this is $f_*(\alpha)$ (I can't find where Strom explicitly states this but this is typically how it's defined). Analogously, as $u\in \tilde{H^k}(Y;A)$ is represented by a map $Y \to K(A, k)$ the pullback $f^*(u)$ to $X$ is represented by the composition $u\circ f\colon X \to K(A,k)$.
Then a representative map for $\langle u, f_*(\alpha) \rangle \in \tilde{H}_{n-k}(Y; A\otimes B)$ can then be given by
$$\begin{align} S^{n + t} &\stackrel{\alpha}{\to} X \wedge K(B,t)\\ &\stackrel{f\wedge id}{\to}Y\wedge K(B,t)\\ &\stackrel{\bar\Delta\wedge id}{\to} Y \wedge Y \wedge K(B,t)\\ &\stackrel{id \wedge u\wedge id}{\to} Y\wedge K(A,k)\wedge K(B,t)\\ &\stackrel{id \wedge c}{\to} Y \wedge K(A\otimes B, k + t) \end{align} $$
according to Strom's definition.
You now have the ingredients you need to come up with a representative for the class $f_*(\langle f^*(u), \alpha \rangle)$. You should do that, and then verify that it agrees with the map that I wrote down (at least up to homotopy).