I am new to the field of algebraic topology and am currently studying Morse homology for a project. I read that given a short exact sequence
$$ 0 \to A \to B \to C \to 0 $$
if we know the orientation of 2 of the 3 vector spaces (or more generally, free R-module), we have an induced orientation on the remaining space. However, I cannot find anywhere how this works exactly. In the case that the chain is split, I suppose we can do this by writing
$$ B\cong A \oplus C $$
But how does this work in the general case?
In particular, I am referring to page 21 of this note on Morse theory, which concerns Morse homology over $\mathbb{Z}$. It would be extra nice if someone can explain to me why, the 3 sequences on page 21, e.g.
$$ 0\to T_x M(c_{k+1},c_k) \to T_xW^s(c_k)\to N_xW^u(c_{k+1})\to 0 $$
are actually short exact sequences. I am not entirely sure what $N_x$ refers to in this case.
Again, since I am new to this area, any help or guidance would be greatly appreciated!