Reference request: Pushforward in Cohomology Given a map $f:X\rightarrow Y$ between top. spaces, there are conditions such that $f$ defines a push-forward or Gysin-map / wrong-way-map in cohomology, that is a homomorphism $f_*: H^*(X)\rightarrow H^*(Y)$.
From what I have read so far, if $f:X\rightarrow Y$ is a fiber bundle of smooth manifolds, then one can define such a map via integration on fibers. More generally, if $f$ is proper, then one can use duality to find a pushforward. And in the algebraic context, proper maps between varieties define a pushforward of the respective Chow - rings, where one needs the notion of a degree of dominant morphisms.
Currently I am interested in the algebraic topology side of things and would like to get enough background knowledge in order to compute pushforwards in simple situations. But the algebraic geometry situation is interesting for me as well.
But I struggle to find a good introduction into the subject. Especially, I don't find any computational examples, which usually help me a lot.
Can someone point me towards some literature?
 A: If both $X$ and $Y$ are oriented manifolds, you can use Poincare duality. If $u\in H^*(X)$, apply the push forward of homology to the poincare dual of $u$ to get some homology class $c$. We define $f_*(u)$ to be Poincare dual to $c$.
Now computing poincare duals is sort of hard. The question “what is the poincare dual of $x+y$ in terms of the poincare duals of $x$ and $y$” leads you to rediscover the universal formal group law in the case that $x$ and $y$ are in $H^2(X)$.
Integrating over the fiber is easier, I think. Bott and Tu defines this operation in their classic book, at least for vector bundles (which is sufficient as I’ll get into in a bit). That’s highly recommended reading in any case. Also Milnor and Stasheff, an other great classic, deduce the gysin sequence for vector bundles, in which the fiber integration appears.
The crucial point is to understand the Thom isomorphism (Which should be covered in both the books I mentioned). You can factor $f:X\rightarrow Y$ through an embedding $i:X\rightarrow E$ for a vector bundle $E\rightarrow Y$. Then there is a nice collapse map $\phi: E\rightarrow X^{\nu_i}$ where this last space is the Thom space of the normal bundle of the embedding $i$. This collapse map just crushes the outside of a tubular neighborhood of $X$ to a point. $f_*$ is just $\phi^*$ composed with the Thom isomorphism of $\nu_i$ on one side, and that of $E$ on the other.
