 
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 fourgenerator
Artin groups, we are able to show that the 5string braid group is the
fundamental group of a compact nonpositively curved space.
Barycentric subdivision subdivides an ncube 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
