Homology of spaces of directed paths in euclidean pattern spaces

Abstract

Let F be a family of subsets of {1,…, n} and let (formula presented) Let XF=Rn\YF. For a vector of positive integers k = (k1,…, k n) let P(XF)k+10 denote the space of monotone paths from 0 = (0,…,0) to k + 1 = (k1 + 1,…, kn + 1) whose interior is contained in XF. The path spaces P(XF)k+10 appear as natural examples in the study of Dijkstra’s PV-model for parallel computations in concurrency theory. We study the topology of P(XF)k+10 by relating it to a subspace arrangement in a product of simplices. This, in particular, leads to a computation of the homology of P(XF)k+10 in terms of certain order complexes associated with the hypergraph F.