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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Discrete Morse theory for manifolds with boundary


Author: Bruno Benedetti
Journal: Trans. Amer. Math. Soc. 364 (2012), 6631-6670
MSC (2010): Primary 57Q10, 57Q15, 05A16, 52B22, 57M25
DOI: https://doi.org/10.1090/S0002-9947-2012-05614-5
Published electronically: April 30, 2012
MathSciNet review: 2958950
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Abstract:

We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities” relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman’s Sphere Theorem. The main corollaries of our work are:

  1. For each $d \ge 3$ and for each $k \ge 0$, there is a PL $d$-sphere on which any discrete Morse function has more than $k$ critical $(d-1)$-cells.

    (This solves a problem by Chari.)

  2. For fixed $d$ and $k$, there are exponentially many combinatorial types of simplicial $d$-manifolds (counted with respect to the number of facets) that admit discrete Morse functions with at most $k$ critical interior $(d-1)$-cells.

    (This connects discrete Morse theory to enumerative combinatorics/ discrete quantum gravity.)

  3. The barycentric subdivision of any simplicial constructible $d$-ball is collapsible.

    (This “almost” solves a problem by Hachimori.)

  4. Every constructible ball collapses onto its boundary minus a facet.

    (This improves a result by the author and Ziegler.)

  5. Any $3$-ball with a knotted spanning edge cannot collapse onto its boundary minus a facet.

    (This strengthens a classical result by Bing and a recent result by the author and Ziegler.)


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Additional Information

Bruno Benedetti
Affiliation: Institute of Mathematics, Freic Universität, Arnimallee 2, 14195 Berlin, Germany
Address at time of publication: Department of Mathematics, Royal Institute of Technology (KTH), Lindstedtsvägen 25, 10044 Stockholm, Sweden
Email: benedetti@math.fu-berlin.de, brunoben@kth.se

Received by editor(s): August 10, 2010
Received by editor(s) in revised form: April 28, 2011
Published electronically: April 30, 2012
Article copyright: © Copyright 2012 American Mathematical Society