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From the topological viewpoint, Morse shellings of finite simplicialcomplexes are {\it pinched} handle decompositions and extend the classicalshellings. We prove that every discrete Morse function on a finite simplicialcomplex induces Morse shellings on its second barycentric subdivision whosecritical tiles-or pinched handles-are in oneto-one correspondence with thecritical faces of the function, preserving the index. The same holds true,given any smooth Morse function on a closed manifold, for any piecewise-lineartriangulation on it after sufficiently many barycentric subdivisions.

Morse shellings out of discrete Morse functions