I'm reading Bredon's Topology and Geometry, VI.9. Duality on compact manifoldss with boundary. In this chapter, I have several questions.
Let $M$ be a conneted orientable $n$-manifold with boundary and $G$ be an abelian group. We shall assume that there is a neighborhood of $\partial M$ in $M$ which is a product $\partial M \times [0,2)$, with $\partial M$ corresponding to $\partial M \times \{0\}$. In p.356 of Bredon, the author consider the follwoing isomorphisms: $$ H^p(M;G) \cong H^p(M-\partial M \times [0,1);G) $$ $$ \cong H_{n-p}(int(M), \partial M \times (0,1);G) $$ $$ \cong H_{n-p}(M,\partial M \times [0,1);G) $$ $$ \cong H_{n-p}(M,\partial M;G) $$ and the author says that, 'By naturality of the cap product, the resulting isomorphism $H^p(M;G) \cong H_{n-p}(M,\partial M;G)$ is the cap product with the orientation class $[M] \in H^n(M,\partial M)$.' Howerver, I have no idea what this sentence means. Why the resulting isomorphism is the cap product with the orientation class?
Let $\Lambda$ be a principal ideal domain, and put $$ \overline{H}^p(\cdot)=H^p(\cdot)/TH^p(\cdot), $$ the torsion free part of the $p$th cohomology group. Since $H_{\cdot}(M;\Lambda)$ is finitely generated, it follows that $\textrm{Ext}(H_{\cdot}(M),\Lambda)$ is all torsion. The author says that 'the universal coefficient theorem gives the isomorphism $\overline{H}^p(M;\Lambda) \to \mathrm{Hom}(\overline{H}_p(M),\Lambda)$. But I think that the resulting isomorphism is $\overline{H}^p(M;\Lambda) \to \overline{\mathrm{Hom}(H_p(M),\Lambda)}$, to a torsion free part of $\mathrm{Hom}(H_p(M),\Lambda)$. Is there an isomorphism between $\overline{\mathrm{Hom}(H_p(M),\Lambda)}$ and $\mathrm{Hom}(\overline{H}_p(M),\Lambda)$?