A note on the combinatorial structure of finite and locally finite simplicial complexes of nonpositive curvature

TL;DR

This paper explores the collapsibility and combinatorial structure of finite and locally finite simplicial complexes with nonpositive curvature, using discrete Morse theory and analyzing their arborescent structures.

Contribution

It introduces a method to analyze collapsibility of systolic complexes and reveals their arborescent structure, extending understanding of nonpositive curvature complexes.

Findings

Systolic complexes are collapsible via discrete Morse theory.

Both CAT(0) and systolic complexes have arborescent structures.

The approach applies to complexes of arbitrary dimension.

Abstract

We investigate the collapsibility of systolic finite simplicial complexes of arbitrary dimension. The main tool we use in the proof is discrete Morse theory. We shall consider a convex subcomplex of the complex and project any simplex of the complex onto a ball around this convex subcomplex. These projections will induce a convenient gradient matching on the complex. Besides we analyze the combinatorial structure of both CAT(0) and systolic locally finite simplicial complexes of arbitrary dimensions. We will show that both such complexes possess an arborescent structure. Along the way we make use of certain well known results regarding systolic geometry.