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PL Morse theory in low dimensions

PL Morse theory in low dimensions AbstractWe 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Geometry de Gruyter

PL Morse theory in low dimensions

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Publisher
de Gruyter
Copyright
© 2023 Walter de Gruyter GmbH, Berlin/Boston
eISSN
1615-715X
DOI
10.1515/advgeom-2022-0027
Publisher site
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Abstract

AbstractWe 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.

Journal

Advances in Geometry – de Gruyter

Published: Jan 1, 2023

Keywords: Morse theory; PL manifold; polytopal complex; 57R70; 57Q99; 52B70; 68Q17

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