On the $m$-dimensional sectional category and induced invariants

Abstract

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.

arXiv comment38 pages. Added Section 7 on Bochner-type theorems, including a lower bound for ${\mathrm{secat}}_1$ in terms of rank of $\ker (p^{\ast })$ and a ${\mathrm{cat}}_m$-version of Oprea's improvement of Bochner's theorem. We prove Proposition 3.5(2) under weaker hypotheses. We introduce $m$-cup length and state Proposition 3.14 using it. Added Proposition 3.21 on ${\mathrm{cat}}_m$ of covers