regular values of PL maps

Question

In the category of smooth manifolds and maps, $y$ is a regular value of $f$ iff the tangent map $df(x)$ is surjective for any $x\in f^{-1}(y)$. Then the preimage $f^{-1}(y)$ is a smooth submanifold.

What is the analogical definition in the category of PL manifolds and PL maps? Some natural definition of "regular values" such that, for a regular value $y$, $f^{-1}(y)$ is a PL submanifold?

It seems to me that if $f: K\to R^n$ is PL, each $x\in f^{-1}(y)$ is contained in a cell of dimension at least $n$ and the restriction of $f$ to each such cell is an affine map of rank $n$, then $f^{-1}(y)$ is a PL submanifold of $M$, but I don't know how to prove it. — Peter Franek, Apr 24 '14 at 18:47

score 2 · Accepted Answer · answered Apr 24 '14 at 19:51

2

This should be in Rourke and Sanderson's book "Introduction to PL topology". Also, take a look at this paper about PL transversality. Once you know what transversality means, if you have a PL map $f: M\to N$ between PL manifolds, then a point $p\in N$ is a regular value of $f$ iff $f$ is transversal to $\{p\}$.

answered Apr 24 '14 at 19:51

Moishe Kohan

97,719

Thanks a lot! Do you know, is my comment above ("It seems to me...") sufficient for $f$ being transversal to $0$? – Peter Franek Apr 25 '14 at 07:05
If by a "cell" you mean an open cell then this condition indeed is sufficient. – Moishe Kohan Apr 25 '14 at 15:31

regular values of PL maps

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