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According to Lawson-Michelsohn's book (p239), the Gysin homomorphism for a continuous map between (compact) manifolds $f:Y\to X$ is defined by setting $$ f_!=PD_X^{-1}\circ f_*\circ PD_Y $$ where $PD$ denote the Poincare duality and $f_*$ is the induced map on homology.

The authors also call $f_!$ `integration along fibers', which makes sense to me if $f$ happens to be a bundle projection map $\pi:E\to X$. But I can hardly understand $f_!$ for an arbitrary continuous map, like the inclusion map $i:X\to E$ of zero section of some vector bundle.

Question 1: Is there a nice way to visualize $f_!$ when $f$ is not a vector bundle map (e.g. an embedding map)?

Question 2: Moreover, is there a more direct definition of $f_!$ without using Poincare duality?

Or, can we make a definition starting from the chain level? For example, if $f=\pi$ is the projection map for an oriented vector bundle, then we do have a well-defined map $\pi_!: \Omega_{vc}^*(E)\to \Omega^*(X)$, see wiki.

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  • $\begingroup$ By Poincaré duality, you can think of $k$-dimensional cohomology classes as being represented by (singular) submanifolds of codimension $k$. Then the Gysin map just takes the image or this submanifold. $\endgroup$
    – Mark Grant
    Jul 13, 2020 at 18:12
  • $\begingroup$ @MarkGrant Thanks! But I'm still confused. So, why not just work with homology groups? what is the good point of using $f_!$ compared to the usual $f_*$? $\endgroup$
    – Hang
    Jul 13, 2020 at 20:35
  • $\begingroup$ Well, there are other good reasons for preferring cohomology over homology (e.g. cup products, cohomology operations) and it's sometimes good to mix these with this extra functoriality for proper maps between manifolds. $\endgroup$
    – Mark Grant
    Jul 14, 2020 at 9:31
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    $\begingroup$ Also check out mathoverflow.net/q/64744/8103 for many different viewpoints on these Gysin maps. $\endgroup$
    – Mark Grant
    Jul 14, 2020 at 9:33

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