Let $M^n$, $n>2$ be a manifold and let $f:D\rightarrow M$ be an embedding of the closed $n-$disk in $M$. Prove or Disprove: $M$ orientable iff $M-f(D)$ is orientable.
$M$ is orientable iff all open sets are, so that takes care of one direction. If $M-f(D)$ is orientable then this should induce an orientation on the boundary $S^{n-1}$ (if we add it back on) and then we can definitely put an orientation on $f(D)$ that agrees with this induced orientation of $S^{n-1}$, giving us a consistent orientation on all of $M$.
But I'm thinking there may be a flaw w that argument, since for example we can remove a $2k$ disk from $\mathbb{R}P^{2k}$ (from the interior of the attached $2k$ cell) and this is homotopy equivalent (I think) to $\mathbb{R}P^{2k-1}$ which is not orientable.
Regarding your concern: note that orientability of a manifold is not a homotopy invariant. E.g. a Mobius strip is not orientable, but it is homotopic to a circle, which is orientable.
(Also, it is even dimension real projective spaces that are not orientable, not odd dimensional ones.)