My question is about p. 69 of Bott Tu. I don't understand how the commutative diagram implies that if $\omega$ is the cohomology class on $M$ representing $S$, then $f^*\omega$ represents $f^{-1}(S)$. I'm also not sure what they mean by "representing" a submanifold—is $\omega$ just the Poincaré dual $\eta_S$, or is it something else?
