Nerves, fibers and homotopy groups

Two theorems are proved. One concerns coverings of a simplicial complex Delta by subcomplexes. It is shown that if every t-wise intersection of these subcomplexes is (k - t + 1)-connected, then for jless than or equal tok there are isomorphisms pi(j)(Delta) congruent to pi(j)(N) of homotopy groups of Delta and of the nerve X of the covering. The other concerns poset maps f : P --> Q. It is shown that if all fibers f(-1)(Q(less than or equal toq)) are k-connected, then f induces isomorphisms of homotopy groups pi(j)(P) congruent to pi(j)(Q), for all jless than or equal tok. (C) 2003 Elsevier Science (USA). All rights reserved.