New Bounds on Distributional Sectional Category and Applications to Distributional Homotopic Distance

Abstract

In this paper, we establish several new bounds for the distributional sectional category ($\mathrm{dsecat}$). We first prove Jauhari's conjecture, thereby establishing a cohomological lower bound for $\mathrm{dsecat}$ with arbitrary coefficients. We then obtain an analogous lower bound with rational coefficients by constructing a natural splitting of the map induced on cohomology by the diagonal inclusion into symmetric powers. As applications, we provide several computations of the distributional sectional category. We also establish multiplicative product and composition inequalities for $\mathrm{dsecat}$. Finally, we apply these results to the distributional homotopic distance recently introduced by Jauhari and Oprea, giving an equivalent formulation in terms of distributed homotopies and deriving new fibration and multiplicative triangle inequalities.