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Transactions of the American Mathematical Society
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
Published electronically: April 30, 2012
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05614-5
PII: S 0002-9947(2012)05614-5
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