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The worst way to collapse a simplex

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Abstract

In general a contractible simplicial complex need not be collapsible. Moreover, there exist complexes which are collapsible but even so admit a collapsing sequence where one “gets stuck”, that is one can choose the collapses in such a way that one arrives at a nontrivial complex which admits no collapsing moves. Here we examine this phenomenon in the case of a simplex. In particular, we characterize all values of n and d so that the n-simplex may collapse to a d-complex from which no further collapses are possible. Equivalently, and in the language of high-dimensional generalizations of trees, we construct hypertrees that are anticollapsible, but not collapsible.

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Acknowledgment

Both authors thank Frank H. Lutz for suggesting this problem and for several helpful comments on an earlier draft of the article.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Davide Lofano & Andrew Newman

Authors
  1. Davide Lofano
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  2. Andrew Newman
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Corresponding author

Correspondence to Andrew Newman.

Additional information

The authors were supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Graduiertenkolleg “Facets of Complexity” (GRK 2434).

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Lofano, D., Newman, A. The worst way to collapse a simplex. Isr. J. Math. 244, 625–647 (2021). https://doi.org/10.1007/s11856-021-2184-3

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  • DOI: https://doi.org/10.1007/s11856-021-2184-3

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