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 graded $\mathbb{Q}$-linear section of the homomorphism induced in cohomology by the symmetric diagonal. 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 distributional homotopic distance. A sequential version of this invariant was recently introduced by Jauhari and Oprea; we give an equivalent formulation in terms of distributed homotopies and derive new fibration and multiplicative triangle inequalities.

arXiv comment31 pages. The paper has been revised with minor edits to improve its clarity and conciseness. In particular, the section on H-spaces has been removed, and several redundant passages have been eliminated to make the presentation more focused