Poincare duality isomorphism problem in the book "characteristic classes" This is the problem from the book, "characteristic classes" written by J.W. Milnor. 
[Problem 11-C] 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 $M$. 

I think that it suffices to show that the right vertical map is sending the fundamental cohomology class of $H^k(N,N-M)$ to the fundamental homology class of $H_n(N) \cong H_n(M)$ because the right vertical map is an isomorphism. However, I can not catch the reason why the right vertical map is an isomorphism. 
Can anybody give me a hint?
Thank you. 

Start picking in the upper left angle the element $u' \otimes \mu_A \in H^k(A,A\setminus M)\otimes H_p(A)$ (using Milnor's notation $u'$ is the dual cohomology class of $M$), then you have to chase this element a little bit. The left "path" $( \downarrow$ and then $\rightarrow)$ is straightforward. You end up with $u'|A\cap \mu_A$.
For the "right" path $(\rightarrow$ and then $\downarrow$ and then $\leftarrow )$ use the so called coherency of the fundamental class $\mu_A$ and Corollary 11.2 in Milnor's Characteristic Classes to prove that you end up in $H^k(N,N\setminus M)\otimes H_p(N,N\setminus M)$ with $th(\nu)\otimes \mu_{N,N\setminus M}$ (work with compactly supported sections of the orientation bundle of $A$, which we know being trivial since $A$ is orientable, for more details see Bredon's Topology and Geometry chapter VI.7). 
Since the vertical right arrow (after choosing the Thom class of the normal bundle as the cohomology class in $H^k(N,N\setminus M)\cong H^k(E(\nu),E(\nu)_0)$) is the so called homological Thom isomorphism (tom Dieck's Algebraic Topology Theorem 18.1.2 page 439), it's an isomorphism so it maps $\mu_{N,N\setminus M}$ to $\mu_M$ (up to a sign and using $H_n(N)\cong H_n(M)$). The conclusion follows at once.
