Cohomological pushforward between closed oriented manifolds ($f^{!}=D_N^{-1}{f}_{\ast }{D}_M$)

For a map between closed oriented manifolds of the same dimension, Poincare duality turns homological pushforward into a degree-preserving map on cohomology. It is adjoint to pullback under the duality pairing: $⟨f^{!}a⌣b,[N]⟩=⟨a⌣f^{\ast }b,[M]⟩$. If $f$ has degree $d$, nondegeneracy gives $f^{!}{f}^{\ast }=d\mathrm{id}$. Maps of different dimensions require the corresponding degree shift.