TITLE: How should one think about pushforward in cohomology?
QUESTION [47 upvotes]: Suppose f:X→Y. If I decorate that first sentence with appropriate adjectives, then I get a pushforward map in cohomology H*(X)→H*(Y).
For example, suppose that X and Y are oriented manifolds, and f is a submersion. Then such a pushforward map exists. In the de Rham picture, we can see this as integrating a form over fibres. In the sheaf cohomology picture, we can see this via the explication of the exceptional inverse image functor.
The question is how else can we think of this pushforward map. I'd be particularly interested in an answer from the algebraic topology point of view, because I'm hoping that such an answer would eludicate the appropriate level of generality in which a pushforward in cohomology exists (perhaps not only answering the question of for which maps f, but also answering the question of in which cohomology theories can we carry out such a construction).

REPLY [9 votes]: Parallel to the OP's two examples, if a cohomology class is defined through intersection with a submanifold (or subvariety with fundamental class in locally finite homology) then the pushforward is defined through intersection with the image of this subvariety. Note how immediate it is to see the change in codimension ( while codimension is constant when taking preimage, which corresponds to the natural map).
As others have said, collapse maps can be used to define pushforwards in general. But having models for pushforwards counts towards understanding a particular cohomology theory geometrically. So for example if you can find a good model for the pushforward of tmf then "you'll win a prize"