I'm trying to better understand how relative homology/cohomology works in order to make use of them for a specific example in mind. Here I'll be working with cohomology over the integers.
Let $A \subset B$ be two topological spaces, $E$ be a fiber bundle over $B$ with projection map $\pi$, and $E|_A$ be the fiber bundle over $A$ obtained from restricting $E$ over the new base space $A$.
I'm interested in both $H^k (B,A)$ and $H^k (E, E|_A)$, and, in particular, whether there is a natural homomorphism from $H^k (B,A)$ to $H^k (E, E|_A)$, possibly induced by $\pi$. So my question is:
Does $\pi$ induce a homomorphism from $H^k (B,A)$ to $H^k (E, E|_A)$? If so, how does one go about finding it, or how is it defined? Does one need stronger conditions on the topological spaces or the fiber bundle to get a meaningful answer?
In general I don't fully understand how relative homology/cohomology works so I don't know where to start in answering this. Any help would be appreciated.