I recently came across the statement “cup product is Poincaré dual to intersection.” I went to look up that result but realized that homological intersection pairing was not even mentioned in any of the books I had on hand (not in Hatcher or in Bott & Tu, for example).
Could someone please recommend a source for this material? I’m also looking for a thorough account of all the relevant ideas and how they fit together—cup product, cap product, Poincaré duality, the de Rham theorem, etc.