Let $f:M\to N$ be a smooth map between two oriented compact smooth manifolds, and denote by $[M]\in H_{\dim M}(M, \mathbb R)$ and $[N]\in H_{\dim N}(N, \mathbb R)$ their fundamental classes. Now for any top de Rham cohomology class $\phi \in H^{\dim N}(N, \mathbb R)$, when does the following relation hold that $$\int_{[M]} f^*\phi = \int_{f_*[M]}\phi$$
Suppose further that $M$ and $N$ are compact. Then we can define a push-forward on the cohomology groups via the Poincare duality $$f_*: H^*(M) \to H^*(N), ~\phi \mapsto PD_M^{-1} \circ f^* \circ PD_N (\phi)$$ where $PD_M: H^*(M)\to H_{n-*}(M)$ denote the Poincare duality. In analogy, when do we have a similar relation as follows: $$\int_{[M]}\phi \cdot f^*\psi=\int_{f_*[M]}f_*(\phi) \cdot \psi$$ where $\phi \in H^*(M)$ and $\psi \in H^*(N)$
Moreover, in what conditions we can further get $$\int_{[M]}\phi \cdot f^*\psi=\int_{[N]}f_*(\phi) \cdot \psi$$