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).
Best Answer
If $X$ and $Y$ are closed manifolds, then you can define the pushforward using Poincaré duality, by pre- and post- composing the map $f_\ast\colon H_\ast(X)\to H_\ast(Y)$ with the appropriate duality isomorphisms. If the manifolds are not compact, then use compactly supported homology and assume $f$ is proper (inverse images of compact sets are compact).
This assumes your manifolds are oriented with respect to whatever homology theory you are using. More generally, you get a pushforward whenever the map $f\colon X\to Y$ is oriented (which roughly amounts to an orientation on the stable normal bundle of $f$) and proper.
I think a good place to read about this is the book "Cohomology Theories" by E. Dyer.
There are other ways to define these pushforward, or Umkehr, maps. You'll find an axiomatic treatment and a survey of the different approaches in this paper of Ralph Cohen and John Klein.