Relating CAT(0) cubical complexes and flag simplicial complexes
Abstract
Given a finite CAT(0) cubical complex, we define a flag simplicial complex associated to it, called the crossing complex. We show that the crossing complex holds much of the combinatorial information of the original cubical complex: for example, hyperplanes in the cubical complex correspond to vertex links in the crossing complex, and the crossing complex is balanced if and only if the cubical complex is cubically balanced. The most significant result is that the sets of $f$vectors of CAT(0) cubical complexes and flag simplicial complexes are equal, up to an invertible linear transformation.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.12517
 Bibcode:
 2020arXiv200412517R
 Keywords:

 Mathematics  Combinatorics;
 52B99 (Primary) 06A07;
 05E45;
 51F99 (Secondary)
 EPrint:
 33 pages, 16 figures