# The worst way to collapse a simplex

## TL;DR

This paper investigates the collapse properties of simplices and hypertrees, revealing cases where collapsibility fails or gets stuck, and explores anticollapsibility in random complexes.

## Contribution

It characterizes when an n-simplex can collapse to a d-complex with no further collapses and constructs hypertrees that are anticollapsible but not collapsible.

## Key findings

- Identifies all (n, d) pairs where an n-simplex collapses to a d-complex with no further collapses
- Constructs hypertrees that are anticollapsible but not collapsible
- Analyzes anticollapsibility phenomena in random simplicial complexes

## 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.