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.
Björner, A. (2003). Nerves, fibers and homotopy groups. Journal of Combinatorial Theory. Series A, 102(1), 88–93. https://doi.org/10.1016/S0097-3165(03)00015-3