On the homotopy theory of complexes associated to metrical-hemisphere complexes

Abstract

In the first section we introduce a certain class of sellular complexes, called metrical- hemisphere complexes (MH-complexes), which generalize oriented matroids, and study their main properties. In Section 2 another cellular complex is associated to each MH-complex; this construction generalizes that given in a preceding work. In Section 3 homotopy theory of the associated complexes is studied by introducing a map ▪ between a certain category of positive paths and the fundamental grupoid. If J is injective then the word problem for the fundamental group is solvable. In Sections 4, 5 higher homotopy groups are considered, giving sufficient criterions for the k(π, 1) property. Also, the results by Deligne concerning the simplicial arrangements is reproved in larger generality. In Section 6 the preceding results are exploited to produce new k(π, 1) spaces associated to arrangement of pseudo-hemispheres.