Summary: BRAIDS, POSETS AND ORTHOSCHEMES
TOM BRADY AND JON MCCAMMOND
Abstract. In this article we study the curvature properties of the order
complex of a graded poset under a metric that we call the "orthoscheme
metric". In addition to other results, we characterize which rank 4
posets have CAT(0) orthoscheme complexes and by applying this the-
orem to standard posets and complexes associated with four-generator
Artin groups, we are able to show that the 5-string braid group is the
fundamental group of a compact nonpositively curved space.
Barycentric subdivision subdivides an n-cube into isometric metric sim-
plices called orthoschemes. We use orthoschemes to turn the order complex
of a graded poset P into a piecewise Euclidean complex K that we call its
orthoscheme complex. Our goal is to investigate the way that combinatorial
properties of P interact with curvature properties of K. More specifically,
we focus on combinatorial configurations in P that we call spindles and
conjecture that they are the only obstructions to K being CAT(0).
Poset Curvature Conjecture. The orthoscheme complex of a bounded
graded poset P is CAT(0) iff P has no short spindles.
One way to view this conjecture is as an attempt to extend to a broader
context the flag condition that tests whether a cube complex is CAT(0). We