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Apparently there are topological and piecewise linear versions of Morse theory. I would like to know of references that treat these topics.

How is a Morse function defined for compact manifolds (with boundary) in the TOP and PL categories?

It is well known that smooth Morse functions always exits for compact smooth manifolds. Are there similar results in the TOP and PL categories?

It is possible to classify closed smooth surfaces via smooth Morse theory. Is there a classification theorem for closed TOP (respestively PL) surfaces via topological (respectively PL) Morse theory?

Thanks

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For TOP Morse function a reference is the classical book of Kirby and Siebenmann "Foundational essays on topological manifolds, smoothings, and triangulations" (1977). The key point is to consider the local standard coordinate charts given by the Morse lemma in the smooth category, and use this to define the TOP Morse functions. These are strictly related to topological handlebody decompositions (so do not exist for non-smoothable topological 4-manifolds). In the PL category you can refer to the link of Daniel Moskovich or google "PL Morse function" (but the TOP approach is not likely to work).

Regarding the second part of the question, you can get such classification once you have proved TOP Morse functions exist! However, the techniques to prove in general that TOP Morse functions exist are typically high-dimensional (dim $\geq 6$). So for surfaces it is likely that proving the existence of TOP Morse functions is equivalent to proving existence of triangulations (which depends on the Schoenflies theorem).

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    $\begingroup$ @Daniele: Thank you for your answer. I want to point out that the existence of triangulations for surfaces does not depend on the Schoenflies theorem (ST). Usually, proofs of the triangulation theorem for surfaces use the ST. This is arguably because once the ST is known the proof becomes simpler. However, using the ST to to prove the existence of triangulations for surfaces is misleading in the sense that the ST fails in dimension 3, but the triangulation theorem holds for 3-manifolds. A proof of the triangualtion theorem for surfaces that does not use the ST is due to E.E. Moise. $\endgroup$
    – Victor
    Commented May 23, 2012 at 6:14
  • $\begingroup$ @Daniele: Moise's proof of the triangulation theorem for surfaces can be found Chapter 7 of "Geometric Topology in Dimensions 2 and 3". $\endgroup$
    – Victor
    Commented May 23, 2012 at 6:15
  • $\begingroup$ Thank you for pointing out this VCF, you're right. Anyway, the proof of Moise is based on the combinatorial Schoenflies theorem (which holds also in $R^3$), since this used in the proof of the approximation Theorem 3 in Chapter 6 in the book of Moise. This approximation theorem is a key ingredient in the proof of the triangulation theorem. $\endgroup$
    – Daniele Zuddas
    Commented May 24, 2012 at 22:52
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Forman's papers and the books by Kozlov and Orlik-Welker that you see cited at Daniel's Wikipedia link are good starting points for the more combinatorial tradition of PL Morse theory.

For more geometric approaches see Bestvina's PL Morse theory and the ancient Piecewise linear critical levels and collapsing by Kearton and Lickorish. Unfortunately, the literature on discrete Morse theory seems to be unaware of the Kearton-Lickorish 1972 paper and the still earlier papers by Kosinski and Kuiper (cited by Kearton and Lickorish) which also constructed some kinds of PL Morse functions. Even if they did it with heavier notation and by less illuminating arguments, their alternative understanding of PL Morse functions is still worth to be aware of.

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    $\begingroup$ There is also Ken Brown's notion of a collapsing scheme from The geometry of rewriting systems: A proof of the Anick-Groves-Squier theorem. Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), 137–163, Math. Sci. Res. Inst. Publ., 23, Springer, New York, 1992, which is an axiomatization of a special case from An infinite-dimensional torsion-free FP_\infty group (with Ross Geoghegan). Invent. Math. 77 (1984), no. 2, 367–381 which is exactly the same as the acyclic matching formulation of Forman's approach but without the assumption of finiteness. $\endgroup$
    – Benjamin Steinberg
    Commented May 14, 2012 at 17:22
  • $\begingroup$ Thanks for these references -- I'm one of these combinatorics people who works with discrete Morse theory and hadn't known about some of them. I suppose it often happens that different areas of math aren't fully aware of each other, though it looks like MathOverflow may help. I'm really appreciative of all the helpful topological insights, references, etc. I've been finding here. Regarding your comment: I wonder if combinatorics people may not all know the precise relationship between being a CW complex and being in PL category, hence the relationship of these different Morse theories. $\endgroup$
    – Patricia Hersh
    Commented May 16, 2012 at 12:26
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    $\begingroup$ One more comment: it's been my impression for awhile now that Bestvina's PL Morse theory is actually quite different from Forman's discrete Morse theory, whereas indeed Brown's notion of collapsing scheme is the same idea from a different viewpoint (with more emphasis on simple homotopy) as Forman's discrete Morse theory. $\endgroup$
    – Patricia Hersh
    Commented May 16, 2012 at 14:24
  • $\begingroup$ Patricia, thanks for your comments. Finite CW complexes with PL attaching maps are polyhedra, i.e. objects of the PL category (not to be confused with polyhedra is the sense of Combinatorics). The relationship of different Morse theories is not simple. Let's start with a discrete Morse function (in the sense of Forman) on a regular CW-complex $X$ such that all cells of $X$ are critical (I guess this is kind of a trivial case of Forman's theory). If $X$ happens to be a PL manifold, the simplicial neighborhoods of the barycenters of cells of $X$ in the second barycentric subdivision of $X$... $\endgroup$
    – Sergey Melikhov
    Commented May 16, 2012 at 14:39
  • $\begingroup$ ... are the handles of a handlebody structure on $X$. Each handle $H=B^n\times B^m$ contains a smaller handle $h=B^n\times (\frac12 B^m)$ near the core of $H$. The given discrete Morse function may be thought of as defined on the barycenters of cells of $X$. One can extend it to a PL function on all of $X$ so that it is constant on the smaller handles $h$, and "vertical" on the collars $H\setminus h$. This will be a Morse function in the sense of Kearton and Lickorish. $\endgroup$
    – Sergey Melikhov
    Commented May 16, 2012 at 14:57
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I think that Daniele and Sergey have, between the two pretty much answered my question. However, I would like to add the following:

  1. Regarding TOP Morse theory, in [Mor1959] M. Morse laid the foundations of the theory of topological non-degenerate functions, and proved the TOP Morse inequalities.

    Also, in [Kui1961] the topological version of the Reeb-Milnor theorem for DIFF manifolds is proven. That is, a TOP n-dimensional closed manifold admiting a TOP Morse function having exactly two non-degenerate critical points is homeomorphic to the n-sphere.

    Another source containing further results for topological manifolds via TOP Morse theory that parallel those obtained for differentiable manifolds is J. Cantwell's paper [Can1967].

    Kirby and Siebenmann's "Foundational essays on topological manifolds, smoothings, and triangulations is freely available here. In section 3 of Essay III (p. 80) they define Morse functions in the DIFF and TOP categories for manifolds possibly with boundary.

    Finally, simple examples of non-differentiable TOP Morse function are easily found. For example, the absolute value function on $\mathbb{R}$, $x\rightarrow |x|$. The origin is a non-degenerate critical point. Also the height function restricted to the double cone (i.e., the space formed by the cones $x^{2}+y^{2}=(z\pm1)^{2}$) has exactly two non-degenerate critical points (the tips of the cones).

  2. Regarding PL Morse theory, J. Harer's slides contain an interesting approach using homology. In particular, a PL Morse function is defined using Betti numbers.

REFERENCES:
[Cant1967] J. Cantwel, Topological non-degenerate functions, Tohoku Math. Journ., 20 (1968), 120-125.

[Kui1961] N.H. Kuiper, A continuous function with two critical points, Bull. Amer. Math. Soc., 67(1961), 281-285.

[Mor1959] M. Morse, Topological non-degenerate functions on a compact manifold $M$, Journal d'Analyse Math., 7 (1959), 189-208.

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There is also a completely unrelated theory the roots of which go to some work of Gromov, see:

MR1621571 (99i:58027) Gershkovich, V.(5-MELB); Rubinstein, H.(5-MELB) Morse theory for Min-type functions. Asian J. Math. 1 (1997), no. 4, 696–715. 58E05 (53C20 53C23 57R70)

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