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.
