I am going through Guillemin and Pollack and have reached some difficulty with orientation. The way it does preimage orientations confuses me, and likewise the problems on the orientation of intersections of submanifolds.
I imagine to compute the orientation of $X \cap Z$, for $X,Z$ submanifolds of $Y$, I'm supposed to use the preimage orientation induced by the inclusion map $X \cap Z \hookrightarrow Y$. Is this correct?
He defines the preimage orientation as follows:
If $f:X \to Y$ is transverse to $Z$, then, the orientation of $S =f^{-1}(Z)$ is given by the following relations:
$$N_x(S; X) \oplus T_x(S) = T_x(X)$$ and $$df_x(N(S;X)) \oplus T_{f(x)}(Z) = T_{f(x)}(Y).$$
So the idea must be, we can determine the orientation of $N(S;X)$ from the second relation, after deducing whether or not $f$ is orientation preserving or reversing, and then plug into the first one to find the orientation of $T_x(S)$? And the orientations "multiply" through (we just need the signs on both sides to be the same)?
Secondly, what is the standard orientation of a point? Oftentimes in problems, $X \cap Z$ is merely a point. For instance, if we take $X$ to be the $x-$axis, $Z$ to be the $y-$axis, and $Y$ to be $\mathbb{R}^2$. How would one go about doing this?
Thanks!