Take the tour ×  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?   share|improve
this question add comment  4 Answers  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 topolo