This is the problem 11-C from the book, "characteristic classes" written by J.W. Milnor.
Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow A$. Let $k = p-n$.
Show that the Poincare duality isomorphism
$\cap \mu_A : H^k(A) \rightarrow H_n(A)$
maps the cohomology class $u^{'}|A$ dual to $M$ to the homology class $(-1)^{nk} i_{*} (\mu_M)$.
[We assume that the normal bundle $v^k$ is oriented so that $\tau_M \oplus v^k$ is orientation preserving isomorphic to $\tau_A|M$.
The proof makes use of the commutative diagram where $N$ is a tubular neighborhood of $M$.
The required isomorhism is easy to prove by chasing the diagram. But what confuses me is that where is the sign $(-1)^{nk}$ from?