Morse theory in TOP and PL categories? 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
 A: I think that Daniele and Sergey have, between the two pretty much answered my question. However, I would like to add the following:


*

*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).

*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.
A: 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) 
A: 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).
A: 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.
