Poincaré dual of Bockstein homomorphism For closed oriented manifold $M^d$, Poincaré duality $H^k(M^d,Z_n) \sim H_{d-k}(M^d,Z_n)$ implies that the Bockstein homomorphism $\beta: H^k(M^d,Z_n) \to H^{k+1}(M^d,Z_n)$, has a dual $\beta^*: H_k(M^d,Z_n) \to H_{k-1}(M^d,Z_n)$.
How to visualize/describe $\beta^*$ in terms of homology?
 A: Poincaré duality takes the cohomological Bockstein homomorphism into the homological Bockstein homomorphism. The latter acts as$$\text{H}_k (X, \mathbb{Z}_n) \to \text{H}_{k - 1}(X, \mathbb{Z})$$—you can further compose this with the reducing map$$\text{H}_{k - 1}(X, \mathbb{Z}) \to \text{H}_{k - 1}(X, \mathbb{Z}_n)$$—and its most classical description is the following. For a given $\gamma \in \text{H}_k(X, \mathbb{Z}_n)$ take a cycle $c \in \gamma$ and then take a $\tilde{c} \in \text{C}_k(X, \mathbb{Z})$ whose projection into $\text{C}_k(X, \mathbb{Z}_n)$ is $c$. Then the boundary $\Delta \tilde{c}$ is not zero (in general), but is divisible by $n$. The quotient $(\Delta \tilde{c})/n$ is a cycle representing $\beta(\gamma)$. The (again, most classical) construction of Poincaré duality is, actually, a construction of an isomorphism between the simplicial chain complex and a barycentric star cochain complex of an oriented closed manifold. Certainly, it takes the Bockstein into the Bockstein.
