= Cohomological pushforward between closed oriented manifolds
{title2=$f^!=D_N^{-1}f_*D_M$}
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: $\langle f^!a\smile b,[N]\rangle=\langle a\smile f^*b,[M]\rangle$. If $f$ has degree $d$, nondegeneracy gives $f^!f^*=d\,\mathrm{id}$. Maps of different dimensions require the corresponding degree shift.
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