Homology of Spaces of Directed Paths in Euclidean Pattern Spaces

Abstract

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

Directed path spaces via discrete vector fields

“…It appears that in many important cases it is not necessary since the differentials vanish by dimensional reasons. This way we reprove here the result of Bjorner and Welker [2], who calculate the homology of "not (k + 1)-equal" configuration spaces on the real line, as well as its generalization due to Meshulam and Raussen [10].…”

Sculptures in Concurrency

“…through the notion of sculptures. Much work has been done in the geometric analysis of Euclidean HDA [12,14,15,21,30,38]; through our equivalences these results are made available for the combinatorial models.…”