# Analogue of singularity theory in other categories

Whitney, Thom, Mather, Arnold and others develoved the singularity theory of smooth maps. Does there exist any analogue of this theory in the category of TOP or PL (or Lipschitz) maps? I mean notions like: stability, hierarchy of singularities, nice dimensions, arising stratification on the source and on the target of a map, according to the germ type of the map at the points, Thom polynomials.

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There has been a considerable amount of work in PL Morse theory which has two flavors: Bestvina's PL Morse theory and Forman's Discrete Morse theory. If you are interested in singularities of $f:X \to Y$ for $Y$ different from the real line, the literature is extremely sparse. But see Cappell and Shaneson's paper on singularities of co-dimension 2 PL embeddings: jstor.org/stable/1971003 –  Vidit Nanda Jul 19 '13 at 23:53
Thank you Vidit. I would need this singularity theory for positive codiension maps (dim target > dim source). But thanks for mentioning PL Morse theory. Can you tell me in two sentences what it is? –  András Szűcs Jul 20 '13 at 7:58
For the PL category, you might be interested in the answers to this question: mathoverflow.net/questions/56597/… –  Daniel Moskovich Jul 20 '13 at 12:53
Thanks Daniel. I can see that MO is really a useful thing. –  András Szűcs Jul 20 '13 at 14:22
@AndrásSzűcs The Cappell-Shaneson paper from my comment deals exclusively with the case dim target = 2 + dim source, which is a (very?) special case of your general interest. I will write a few words on PL Morse theory as an answer because it is difficult to format math in comments. –  Vidit Nanda Jul 20 '13 at 14:37

A brief account of PL Morse theory has been requested by Andras in the comments, so I am writing it down here. Note that this does not address the main question on PL singularity theory. Note also that a much better account of PL Morse theory can be found in Mladen Bestvina's notes.

Let $K$ be a polyhedral cell complex and denote the attaching map of each $n$-simplex $\sigma$ of $K$ by $\phi_\sigma:\Delta^n \to K$ where $\Delta^n$ is the standard $n$-simplex.

Def. A function $f:K \to \mathbb{R}$ is PL Morse if (a) $f\circ \phi_\sigma:\Delta^n \to \mathbb{R}$ is constant only for $n=0$, affine for $n \geq 1$, and (b) the image under $f$ of the vertices forms a discrete subset of $\mathbb{R}$.

This definition sets things up so that the only candidates for critical points of $f:K \to \mathbb{R}$ are the vertices of $K$. For each vertex $v$, the descending link $\text{Lk}^-_f(v)$ consists of all those simplices $\sigma$ not containing $v$ such that $v \cup \sigma =: \sigma'$ is a simplex of $K$ and $f|_{\sigma'}$ attains its maximum on $v$. The descending link plays the role of stable manifold in PL Morse theory. Here's the main Theorem:

Thm. If $I = [a,b]$ is an interval so that the only vertex in $f^{-1}(I)$ is $v$ with $f(v) = b$, then $f^{-1}(I)$ is homotopy equivalent rel $f^{-1}(a)$ to $f^{-1}(a)$ attached with the cone on $\text{Lk}^-_f(v)$.

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Thanks, clear. Can one also define analogues of birth death singularites? (As the non-Morse singularities in typical one parameter families.) And then also the higher singularities as well? –  András Szűcs Jul 20 '13 at 20:28
@AndrásSzűcs I don't think anyone has systematically studied "PL Cerf theory": in general I don't see why homotopies $F_t: K \to \mathbb{R}$ of locally affine maps should be locally affine for almost all $t$ unless you are careful. There is some progress in corresponding questions for discrete Morse theory (i.e., Forman's rather than Bestvina's) which you can find in my MO answer to Daniel Moskovich's question here: mathoverflow.net/a/105469/18263 –  Vidit Nanda Jul 20 '13 at 21:28