Nerves, fibers and homotopy groups

Two theorems are proved. One concerns coverings of a simplicial complex D by subcomplexes. It is shown that if every t-wise intersection of these subcomplexes is ðk t þ 1Þ-connected, then for jpk there are isomorphisms pjðDÞDpjðNÞ of homotopy groups of D and of the nerve N of the covering. The other concerns poset maps f : P-Q: It is shown that if all fibers f ðQpqÞ are kconnected, then f induces isomorphisms of homotopy groups pjðPÞDpjðQÞ; for all jpk: r 2003 Elsevier Science (USA). All rights reserved.