Romain Grunert, Wolfgang Kühnel, and Günter Rote:

PL Morse theory in low dimensions

Manuscript, December 2019, 24 pages, arXiv:1912.05054 [math.GT], submitted for publication.  →BibTeX

Abstract

We discuss a piecewise linear (PL) analogue of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level, it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong regularity are presented. In particular we show that in low dimensions d≤4 a homologically regular point on a PL d-manifold is always strongly regular. Examples show that this fails to hold in higher dimensions d≥5. One of our constructions involves an 8-vertex embedding of the dunce hat into a polytopal 4-sphere with 8 vertices such that a regular neighborhood is Mazur's contractible 4-manifold.

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Last update: December 12, 2019.