Let $M$ be a $4$- dimensional oriented manifold and let D be a 2-dimensional submanifold. Is it true that any $\phi \in \text{Diff}^+(M)$(the set of orientation preserving diffeomorphisms) preserves homological intersection i.e $[D] \cdot [D] = \phi_{*}[D] \cdot \phi_{*}[D]$.

  • $\begingroup$ Thank you. Just wanted to further know how do you go about showing the claim that $[\phi_{*}(D)] \cdot [\phi_{*}(D)] = \phi_{*}([D] \cdot [D])$? $\endgroup$ – J-holo curve Dec 20 '18 at 1:33
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    $\begingroup$ Poincare duality is natural for orientation preserving diffeomorphisms, and the intersection product is Poincare dual to the cup product. Alternatively; pick transverse representatives and literally push them forward, and the intersection of the pushed forward cycles will be the pushforward of the intersection. $\endgroup$ – Mike Miller Dec 20 '18 at 1:59

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