Krdytkyu

Let $X,Y$ be (not necessarily compact) orientable manifolds without boundaries of dimension $m$ and $n$. Let $f : X rightarrow Y$ be a proper map. Assume that all the cohomologies have coefficients in $mathbb{Q}$. We could define the following pullback maps

$$f^* : H^*(Y) rightarrow H^*(X), f^*_c : H^*_c(Y) rightarrow H^*_c(X)$$

We also have the following maps defined by their poincare duals

$$ f_{!,c} : H^*_c(X) rightarrow H^{*+(n-m)}_c(Y), f_! : H^*(X) rightarrow H^{*+(n-m)}(Y) $$

I am interested in knowing the projection formulas that are available in this setup. A little bit of googling up gives the following link

https://mathoverflow.net/questions/67228/where-do-all-these-projection-formulas-come-from

and

https://mathoverflow.net/questions/18799/ubiquity-of-the-push-pull-formula

which seems useful but they are not the kind of equality I am interested in. I am wondering if the following is true for $alpha in H^*_c(Y), beta in H^*(X)$.

$$f_!(f^*_c(alpha) cup beta) = alpha cup f_!(beta)$$

Q1. Are there other kinds of projection formulas available?

Q2. Which of them are poincare dual or related to the others?

It would be extremely helpful if the answers contain references if not the proofs themselves. I am not sure if the notations I have used are the correct ones. Please feel free to edit accordingly.

Thanks!