I'm trying to understand the Case $1$ of Poincaré Duality Theorem (Massey p.$361$).
Theorem: Let $M$ be an $n-$oriented manifold and $G$ an arbitrary abelian group. Then the homomorphism $$P : H_c^q(M;G) \longmapsto H_{n-q}(M;G)$$ is an isomorphism for all q.
The proof goes by step. In particular in the first one we have $M = \mathbb{R}^n$ and we reduce the theorem to prove that $H^n(\mathbb{R}^n,\mathbb{R}^n-B;G) \longmapsto H_{0}(\mathbb{R}^n;G)$ given by $x \to x \frown \mu_B$ is an isomorphism, where $B$ is a closed ball in $\mathbb{R}^n$.
The argument is the following :
Now $\mu_B$ is a generator of the infinite cyclic group $H_n(\mathbb{R}^n,\mathbb{R}^n-B,\mathbb{Z})$. We will complete the proof using the following relation: $$\varepsilon_*(x \frown \mu_B) = \langle x,\mu_B \rangle$$
Since $\mathbb{R}^n$ is arcwise connected $\varepsilon_* : H_0(\mathbb{R}^n;G) \longmapsto G$ is an isomorphism. Moreover, by the universal coefficient theorem for cohomology the homomorphism $$\alpha : H^n(\mathbb{R}^n,\mathbb{R}^n-B,G) \longmapsto \text{Hom}(H_n(\mathbb{R}^n,\mathbb{R}^n-B,Z),G)$$ is also an isomorphism (where $\alpha(\varphi)[c] = \varphi(c)$. Using the definition of $\alpha$ in terms of the scalar product, the desidered conclusion follows.
I have two questions regarding this proof:
$1.$ Why $\mu_B$ is a generator of $H_n(\mathbb{R}^n,\mathbb{R}^n-B,\mathbb{Z}) \simeq \mathbb{Z}?$ In the definition of orientation we don't require that the class $\mu_B$ which restricts to local orientations $\mu_y$ for every $y \in B$ is a generator. Am I wrong?
$2.$ How the thesis should follow directly from the definition of $\alpha?$
My thoughts: I could try to use the formula $\langle \tau, \beta \frown \gamma \rangle = \langle \alpha \smile \beta, \gamma \rangle$ with $\tau \in H^0,\beta = \mu_{B}^{*}$(where $\mu_B^*$ is the cochain that defines the isomorphism $\alpha$) and $\gamma = \mu_B$ but I don't know if this is enough. Could I take $\tau$ as the neutral element in $H^0$ in order to achieve that the cap product with $\mu_B$ is the composition of two isomorphism?
Any help, answer or reference would be appreciated.
Edit:
Problem $1$ has been solved, one can require that for specific compact sets, such as closed balls in $\mathbb{R}^n$ $\mu_B$ is a generator thanks to the fact that in a connected manifolds orientations must equal up to sign.
As far as concerns $2$. my doubt remains. I tried to consult Hatcher but it seems that the proof of Hatcher is for $G = \mathbb{Z}$ (correct if I'm wrong otherwise I didn't understand fully universal coefficient Theorem) but I think his method can't be replied since if the group $G$ is not cyclic I can't prove that the cap product is an isomorphism since a generator of $H_n(\Delta^n,\partial \Delta^n)$ is represented by a cocycle which takes value $1$ on $\Delta^n$.