I do not understand the highlighted part of the paragraph given below:
Could anyone explain it for me please?
2026-03-25 06:12:53.1774419173
A difficulty in understanding a part of a paragraph in Guillemin & Pollack p.60
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For their example map $f$, we have $f^{-1}(Z) = \partial H^2$, which is the $x$-axis in the plane. This is boundaryless, so $\partial f^{-1}(Z)$ is empty. On the other hand, $f^{-1}(Z) \cap \partial H^2 = \partial H^2$ which is the $x$-axis and not empty. Hence $\partial f^{-1}(Z) = f^{-1}(Z) \cap \partial H^2$ is false in this example. Since you would rather have the boundary of the preimage be the part of the preimage that lives in the boundary, you will have to make additional assumptions above and beyond transversality.