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By Poicaré Poincaré duality, there is an isomorphism

$H_2(M, \mathbb{Z}) \cong H^2(M, \mathbb{Z})$.

Now let $PD(a) \in H^2(M, \mathbb{Z})$ be the Poincaré dual of $a$. Since $H^2(M, \mathbb{Z})$ classifies line bundles on $M$, there exists a line bundle $L$ such that $c_1(L)=PD(a)$. Take a general smooth section of $L$. Then its zero set is a smoothly embedded oriented surface $\Sigma \subset M$ such that its fundamental class $[\Sigma]$ is equal to $a$.

See Donaldson-Kronheimer ["The geometry of 4-manifolds", Chapter 1] for more details.

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By Poicaré duality, there is an isomorphism

$H_2(M, \mathbb{Z}) \cong H^2(M, \mathbb{Z})$.

Now let $PD(a) \in H^2(M, \mathbb{Z})$ be the Poincaré dual of $a$. Since $H^2(M, \mathbb{Z})$ classifies line bundles on $M$, there exists a line bundle $L$ such that $c_1(L)=PD(a)$. Take a general smooth section of $L$. Then its zero set is a smoothly embedded oriented surface $\Sigma \subset M$ such that its fundamental class $[\Sigma]$ is equal to $a$.

See Donaldson-Kronheimer ["The geometry of 4-manifolds", Chapter 1] for more details.