Nerves, fibers and homotopy groups
A much used result of \textit{D. Quillen} [Adv. Math. 28, 101-128 (1978; Zbl 0388.55007)] is that a map of posets \(f:P\to Q\) with contractible fiber is a homotopy equivalence (simplicial complexes are associated to posets in the usual way). The first main result of the paper under review extends this by only requiring the map to be \(k\)-connected and concluding that \(f\) induces an isomorphism on homotopy groups through dimension \(k\). The second result is a similar extension of \textit{K. Borsuk}'s theorem on nerves [Fundam. Math. 35, 217-234 (1948; Zbl 0032.12303)] that a simplicial complex \(P\) is homotopy equivalent to the nerve of any cover of \(P\) by subcomplexes with all intersections either empty or contractible. Namely, if the condition on intersections is weakened to connectivity through a range (depending on the number of subcomplexes that are being intersected), then there is a map from \(P\) to the nerve of the cover that induces an isomorphism on homotopy groups through a range. (Quillen's and Borsuk's results follow from these extensions by the Whitehead theorem.).
scientific article; zbMATH DE number 1917909