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Assume $M$ is a smooth connected manifold with $H^k(M)$ finite, $k\leq \dim M$. Is it true that the Kronecker pairing $$ \langle,\rangle:H^k(M)\times H_k(M)\longrightarrow \mathbb{Z} $$ $$ \langle[\varphi],[\Sigma]\rangle=\varphi(\Sigma) $$ is trivial?

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1 Answer 1

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Hint : a morphism $\Bbb Z/m \Bbb Z \to \Bbb Z$ is always zero.

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