Assume that $M$ is $n$-dimensional, compact, connected and oriented with boundary manifold. Show that if $H_{k-1}(M,\partial M, Z)$ is torsion-free, then $H_{n-k}(M, Z)$ is torsion-free.
This question is a my PhD. qualification exam question. I didn't solve this question in exam today. Can you help me?
By Poincaré-Lefschetz duality, $H_{k-1}(M,\partial M;\mathbb{Z})$ is isomorphic to $H^{n-k+1}(M;\mathbb{Z})$. The universal coefficient theorem gives that $H^{n-k+1}(M;\mathbb{Z})$ is isomorphic to $\operatorname{Hom}(H_{n-k+1}(M;\mathbb{Z}),\mathbb{Z})\oplus\operatorname{Ext}(H_{n-k}(M;\mathbb{Z}),\mathbb{Z})$. If $H_{n-k}(M;\mathbb{Z})$ had torsion, then this would appear in the $\operatorname{Ext}$ group, leading to torsion in $H_{k-1}(M,\partial M;\mathbb{Z})$.