Directed path spaces via discrete vector fields

Abstract

Let K be an arbitrary semi-cubical set that can be embedded in a standard cube. Using Discrete Morse Theory, we construct a CW-complex that is homotopy equivalent to the space P(K ) w v of directed paths between two given vertices v, w of K . In many cases, this construction is minimal: the cells of the constructed CW-complex are in 1-1 correspondence with the generators of the homology of P(K ) w v .

Sculptures in Concurrency

“…Technically, geometric sculptures are Euclidean cubical complexes; rewriting a proof in [38] we show that such complexes are precisely (combinatorial) sculptures. In other words, a HDA is Euclidean iff it can be sculpted, so that the geometric models for concurrency [12,13] are closely related to the combinatorial ones [25,32], Fig.…”

“…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.…”

Spaces of directed paths on pre-cubical sets II

“…The problem of calculating of the homotopy types of d-path spaces between two vertices of a -set was studied in several papers, eg. [10,11,16,17]. All these results work only for special classes of -sets, like Euclidean complexes or proper -sets, i.e., ones that their triangulations are simplicial complexes.…”