Let $M$ be an closed manifold, $D$ a disc inside it. As far as I understand, in orientable case the only difference between the homology (over a given field) of $M$ and $M \setminus D$ is one more dimension in $n-1$ homology: although Mayer-Vietoris sequence gives also possibility of one less dimension in $n$ homology, orientability kills it.
Does this later possibility really appear in non-orientable case? Then the deletion of disc should change orientability, is it possible?
Hint 1:
"If you delete a disk from a non-orientable manifold can you obtain an orientable manifold with boundary?"
should be the same as
"Can you obtain a non-orientable manifold from a connected sum of an orientable manifold and a sphere?"
(See connected sums of closed orientable manifold is orientable)
Hint 2:
A n-dimensional manifold is non-orientable if it contains a homeomorphic image of the space formed by taking the direct product of a (n-1)-dimensional ball B and the unit interval [0,1] and gluing the ball B×{0} at one end to the ball B×{1} at other end with a single reflection. (see http://en.wikipedia.org/wiki/Orientability)
If $M$ contains such a space, then also $M\setminus D$ contains such a space.
For what concern top homology, $H_{n}(M,\mathbb{Z})=0$ for non-orientable manifolds, so it cannot 'decrease'.
