Homotopy types of finite étale spaces and generalized inflations

Abstract

Inflation of a simplicial complex $K$ is a construction well known in combinatorial topology. It replaces each vertex $i$ of $K$ with a finite number $n_i$ of its copies, and each simplex $\{i_0,\ldots,i_k\}$ with $n_{i_0}n_{i_1}\cdots n_{i_k}$ many copies so that the collection of vertex-copies is spanned by a simplex in the inflation if and only if their originals were spanned by a simplex in the original complex. The celebrated poset fiber theorem of Björner, Wachs, and Welker describes the homotopy type of such inflation in terms of homotopy types of $K$ and its links. In the current paper, we introduce more general inflations over simplicial posets: we replace each simplex with an arbitrary finite set of copies. The way how these sets patch together is specified by a commutative diagram, or, equivalently, a sheaf on the corresponding finite topology. The generalized inflation can be understood as étale space of such sheaf. We prove that, whenever this inflation sheaf is flabby, the poset fiber theorem still applies. We prove all results similar to those known for vertex inflations. We also cover the previous result of the first author about homotopy types of clique complexes of multigraphs.