Two theorems are proved. One concerns coverings of a simplicial complex 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 ≅ πj(N) of homotopy groups of and of the nerve N of the covering. The other 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. © 2003 Elsevier Science (USA ). All rights reserved.
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