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Let $M$ be an oriented compact smooth manifold of dimension $n$. Let $[M]$ be the fundamental class of $M$, that is, $[M]\in H_n(M, \mathbb Z)$. Then, the Poincare duality map is the isomorphism given by $$ H^p(M, \mathbb Z)\to H_{n-p}(M, \mathbb Z), \quad [\alpha]\mapsto[\alpha]\frown[M]. $$

Is the inverse of the above map known and can it be explicitly defined?

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1 Answer 1

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Let $|-\cap^{trans}-|$ denote the operation of counting signed intersections of transverse representatives of embedded simplicial complexes of complementary dimension. The inverse map is given by sending a class $[\sigma]$ to the adjoint of its intersection pairing, i.e. the cohomology class of the cocyle $\rho \rightarrow |\rho \cap^{trans} \sigma|$. For more information, see section 3.6 of Whitney Stratified Chains and Cochains.

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