Distributional category of manifolds
Ekansh Jauhari
TL;DR
This work develops the distributional Lusternik–Schnirelmann category $\mathsf{dcat}$ as a robust lower bound for the classical category $\mathsf{cat}$ and provides broad, obstruction-theoretic criteria under which $\mathsf{dcat}$ attains the maximal value $\mathsf{cat}$. The authors derive sufficient cap-property conditions for closed essential manifolds with torsion-free fundamental groups, extend these ideas to connected sums and generalized summands, and obtain extensive new computations of $\mathsf{dcat}$ in low dimensions, including $3$- and $4$-manifolds, substantive generalized connected-sum results, and Alexandrov-space extensions. They also initiate a program to transfer these results to closed Alexandrov spaces, establishing partial connected-sum formulas and identifying ample classes where $\mathsf{dcat}$ equals the topological dimension. The paper further explores an analog of Rudyak’s conjecture for $\mathsf{dcat}$ in several cases, and demonstrates concrete instances where $\mathsf{dcat}(X)=\mathsf{cat}(X)$, enhancing the applicability of obstruction-theory techniques to modern topological invariants with geometric and group-theoretic connections.
Abstract
Recently, a new homotopy invariant of metric spaces, called the distributional LS-category, was defined, which provides a lower bound to the classical LS-category. In this paper, we obtain several sufficient conditions for the distributional LS-category (dcat) of a closed manifold to be maximum, i.e., equal to its classical LS-category (cat). These give us many new computations of dcat, especially for some essential manifolds and (generalized) connected sums. In the process, we also determine the cat of closed 3-manifolds having torsion-free fundamental groups and some closed geometrically decomposable 4-manifolds. Finally, we extend some of our results to closed Alexandrov spaces with curvature bounded below and discuss their cat and dcat in dimension 3.