Nerves, fibers and homotopy groups

Two theorems are proved. One concerns coverings of a simplicial complex Δ by subcomplexes. It is shown that if every t-wise intersection of these subcomplexes is (k−t+1)-connected, then for j⩽k there are isomorphisms πj(Δ)≅πj(N) of homotopy groups of Δ and of the nerve N of the covering. her concerns poset maps f : P→Q. It is shown that if all fibers f−1(Q⩽q) are k-connected, then f induces isomorphisms of homotopy groups πj(P)≅πj(Q), for all j⩽k.