The worst way to collapse a simplex Davide Lofano * Andrew Newman * Institut f¨ ur Mathematik, TU Berlin November 24, 2021 Abstract In general a contractible 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. Furthermore we examine anticollapsibility in random simplicial complexes. 1 Introduction A standard notion in computational topology is that of collapsibility, first introduced by White- head [Whi39] as a combinatorial version of contractibility. For a simplicial complex X,a nonempty face τ of X is said to be free provided that it has only one proper coface; an elementary collapse of X is the process of removing some free face τ and its unique proper coface σ. A simplicial complex is said to be collapsible, if there exists a sequence of elementary collapses that reduce the complex to a single vertex. Now an elementary collapse is a homotopy equivalence, and so if a simplicial complex is collapsible then it is contractible. On the other hand, the converse is not true in general. For example, the dunce hat [Zee64], which we will discuss in more detail below, is a 2-dimensional contractible complex with triangulations on 8 vertices which are not collapsible. Moreover, this example is vertex minimal, as a result of [BD05] shows that for simplicial complexes on 7 or fewer vertices contractibility and collapsibility are equivalent. Our first main result builds on this to fully characterize n and d so that there exists d-dimensional simplicial complexes on n vertices which are contractible, but for which not a single elementary collapse is possible. Theorem 1.1. For every n ≥ 8 and d with d/ ∈{1,n − 3,n − 2,n − 1} there exists a contractible d-complex on n vertices with no free faces. Moreover this is best possible, for n ≤ 7 or d ∈ {1,n − 3,n − 2,n − 1} every contractible complex has a free face. The constructions we produce for Theorem 1.1 are related to another question of collapsibil- ity. While collapsibility and contractibility are not equivalent, more subtlety there even exists simplicial complexes which are collapsible, but for which it is possible to choose a sequence of elementary collapses after which one gets stuck at a nontrivial complex that is not collapsible. * Supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Graduiertenkolleg “Facets of Complexity” (GRK 2434). 1 arXiv:1905.07329v2 [math.CO] 13 Jun 2019