arXiv subjects
Publications and source records attributed to Sutirtha Datta.
We introduce and study discrete analogs of Scott's and Murillo-Wu's topological complexity of maps. We prove that these discrete analogs are contiguity invariants and are, in fact, equivalent. Furthermore, we establish the fundamental theoretical properties and computational aspects of the discrete topological complexity of simplicial maps.
In this paper, we systematically study the $m$-dimensional sectional category of a fibration, introduced by Schwarz as an approximating invariant for the sectional category. We develop the basic theory of this invariant, establish its fundamental properties, and show how it gives rise to a hierarchy of induced invariants, including the $m$-dimensional Lusternik--Schnirelmann category, the $m$-topological complexity, and the $m$-homotopic distance between maps. We further investigate the relationships between these $m$-dimensional invariants and their classical analogues, present a variety of examples in which these invariants are computed, and illustrate when they agree with or differ from their classical counterparts. We also introduce the notion of $m$-cohomological distance and study its interaction with the $m$-homotopic distance. Finally, we prove a Bochner-type theorem for $\mathrm{secat}_1$, extending the corresponding theorem of Oprea and Strom for $\mathrm{cat}_1$. We also establish a $\mathrm{cat}_m$ version of Oprea's improvement of Bochner's theorem.
We introduce and study a parametrized analogue of the directed topological complexity, originally developed by Goubault, Farber, and Sagnier. We establish the fibrewise basic dihomotopy invariance of directed parametrized topological complexity and explore its relationship with the parametrized topological complexity. In addition, we introduce the concept of the directed Lusternik-Schnirelmann (LS) category, prove its basic dihomotopy invariance, and investigate its connections with both directed topological complexity and directed parametrized topological complexity. We further investigate additional properties of our invariant and examine its connections with several other invariants that arise naturally in the context of topological robotics. Moreover, we compute the directed parametrized topological complexity of the Hopf fibrations and the Fadell-Neuwirth fibrations having specific directed fibration structures.
We study the higher (sequential) topological complexity, a numerical homotopy invariant for the planar polygon spaces. For these spaces with a small genetic codes and dimension $m$, Davis showed that their topological complexity is either $2m$ or $2m+1$. We extend these bounds to the setting of higher topological complexity. In particular, when $m$ is power of $2$, we show that the $k$-th higher topological complexity of these spaces is either $km$ or $km+1.$