Poset fiber theorems

Abstract

Suppose that f : P → Q f:P \to Q is a poset map whose fibers f − 1 ( Q ≤ q ) f^{-1}(Q_{\le q}) are sufficiently well connected. Our main result is a formula expressing the homotopy type of P P in terms of Q Q and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences or special cases. Homology, Cohen-Macaulay, and equivariant versions are given, and some applications are discussed.