I started reading about the theory of surgery and looked at the wikipedia page (https://en.wikipedia.org/wiki/Surgery_theory) to see if it contained any interesting examples. The section "Effects on homotopy groups and the comparison with cell attachment" contains an interesting statement.
He begins by describing surgery as attaching a $ (p + 1) $-cell and remove a $ q $-cell, as I understand it. They then say that the need to remove the $ q $-cell can be understood as an effect of the Poincaré duality. Previously, I understood that this detachment process was just playing a numbers game to match the dimensions. I am very interested in this subject. What does it really mean? How does the duality of Poincaré explain the need to detach a $ q $-cell?
Thank you for any help