I am reading Milnor's Topology from a Differentiable Viewpoint, which is very nice. I am reading about regular points versus critical points, etc. I ran across some notation that I was not very clear about. On page 11, as part of the proof of Brown's theorem, he references a particular operator, and I was not clear on what the notation was actually referring to. if anyone can explain, it would really help.
So he is talking about an manifold $M \subset R^k$,$N \subset R^m$ and a linear map $L: R^k \rightarrow R^{m-n}$ that is nonsingular on the subspace $\mathfrak{R} \subset TM_x \subset R^k$.
This is where it gets confusing for me:
Now define $$ F: M \rightarrow N \times R^{m-n} $$
by $F(\xi) = (f(\xi), L(\xi))$ The derivative $dF_x$ is clearly given by the formula
$$ dF_x = (df_x(\nu), L(\nu)) $$
I was trying to understand what this mapping $F: M \rightarrow N \times R^{m-n}$ means, and this reference to $F(\xi) = (f(\xi), L(\xi))$. Is $F$ the cartesian product of all points in N with all vectors in $R^{m-n}$? What would be the use of that? Also, this $F(\xi)$ notation, what does it mean that I have $(f(\xi), L(\xi))$. Like what does that tuple represent, if both of those are mappings to different sized spaces?
So I think I am missing what the intent of the notation is.
Yes, it's the cartesian product. And the use is, very likely, that, for the combined map, dimension counting is easy and the implicit or inverse function theorem can be applied.