TITLE: Morse theory in TOP and PL categories?
QUESTION [17 upvotes]: 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

REPLY [5 votes]: 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.