What is the essence of ``the naturality of the cap product''?

Question

Associated to a continuous map $f : X → Y$, there are natural pushforward and pullback maps on homology and cohomology, respectively, denoted $f_∗ : H_∗(X) → H_∗(Y)$ and $f^*: H^* (Y) → H^* (X)$. These are related by the projection formula, also called ``the naturality of the cap product'': $$f_∗(f^∗ c \cap σ) = c \cap f_∗ σ.$$

My questions are that:

1. what is the essence of this ``the naturality of the cap product''?

2. what are the uses of this ``the naturality of the cap product''?

Suppose it can be used in the proof of Alexander-Lefschetz-Poincaré Duality. What are the essence behind?

Answer 1

1. One way to think about the cap product, which is a pairing $$ \cap\colon H_{k+l}(X)\otimes H^k(X) \rightarrow H_l(X) $$ is via its adjoint which is a homomorphism $$ \alpha_X \colon H_{k+l}(X)\rightarrow \hom(H^k(X),H_l(X)). $$ Notice that both sides of the equation are covariant functors in $X$ and naturality of the cup product precisely means that $\alpha_X$ is a natural transformation.

2. A very important application of this naturality statement is the following: Let $$f\colon M \rightarrow N$$ be a continous map of degree $d$ between closed, connected and oriented manifolds of dimension $n$. The Poincaré duality isomorphism is defined as the "capping with the fundamental class of a manifold": $$ PD_M = \_\cap [M]\colon H^k(M) \rightarrow H_{n-k}(M).$$ Let us apply naturality of the cap product to our map $f$. For $\phi\in H^k(N)$: $$f_*(f^*(\phi)\cap [M]) = \phi \cap d\cdot [N] =d\cdot (\phi\cap [N]). $$ If $f^!\colon H_k(N)\rightarrow H_k(M)$ denotes the "Umkehr map" $PD_M\circ f^* \circ PD^{-1}_N$, the the above shows that $f_* \circ f^! \colon H_k(N)\rightarrow H_k(N)$ is multiplication by $d$, which can be useful. In particular, if $d=1$, then $f_*$ is split surjective. Also, using the Theorem of Whitehead, a map of degree $1$ from a sphere to a closed and simply connected manifold is a homotopy equivalence.