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Contiguity distance between simplicial maps
(2023) Borat, Ayse; Pamuk, Mehmetcik; Vergılı, Tane
For simplicial complexes and simplicial maps, the notion of being in the same contiguity class is defined as the discrete version of homotopy. In this paper, we study the contiguity distance, SD , between two simplicial maps adapted from the homotopic distance. In particular, we show that simplicial versions of LS -category and topological complexity are particular cases of this more general notion. Moreover, we present the behaviour of SD under the barycentric subdivision, and its relation with strong collapsibility of a simplicial complex.
Higher analogues of discrete topological complexity
(Springer-Verlag Italia Srl, 2024) Alabay, Hilal; Borat, Ayse; Cihangirli, Esra; Erdal, Esma Dirican
In this paper, we introduce the nth discrete topological complexity and study its properties such as its relation with simplicial Lusternik-Schnirelmann category and how the higher dimensions of discrete topological complexity relate with each other. Moreover, we find a lower bound of n-th discrete topological complexity which is given by the nth usual topological complexity of the geometric realisation of that complex. Furthermore, we give an example for the strict case of that lower bound.