# algebraic topology – Variety surgery and Poincaré duality

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