Abstract
We discuss a PL analog 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 dimensions d ≤ 4 a homologically regular point on a PL d-manifold is always strongly regular. Examples show that this fails in higher dimensions d ≥ 5. One of our constructions involves an embedding of the dunce hat into 4-space and Mazur’s contractible 4-manifold. Finally, decidability questions in this context are discussed.
Communicated by: M. Joswig
Acknowledgements
This research was supported by the Deutsche Forschungsgemeinschaft (DFG — German Research Foundation) — Project-ID 195170736 — as part of the Collaborative Research Center TRR 109, Discretization in Geometry and Dynamics.
We thank Benjamin Burton for checking the topology of the tubular neighborhood M in Section 8 with the Regina software.
References
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