LS-category and topological complexity of real torus manifolds and Dold manifolds of real torus type
Koushik Brahma, Navnath Daundkar, Soumen Sarkar
TL;DR
This work studies Lusternik–Schnirelmann category and (symmetric) topological complexity for real torus manifolds, including generalized real Bott manifolds and small covers, and for Dold manifolds of real torus type. It develops sharp cohomological methods based on zero-divisors cup-length and equivariant categories, yielding exact or near-exact bounds for TC and, in many cases, the precise values of these invariants, by exploiting the orbit-space combinatorics and the mod-$2$ cohomology ring presentations. The authors provide a simplified presentation of the cohomology rings for generalized real Bott towers, prove that $\mathrm{cat}(M)=n+1$ under suitable boundary conditions on the orbit space, and derive improved TC bounds when the simplicial factors contain odd dimensions. They further compute the $\mathbb{Z}_2$-equivariant LS-category for small covers with path-connected fixed points, determine $\mathrm{cat}$ and $\mathrm{TC}$ for Dold manifolds of real torus type, and establish sharp bounds for the symmetric topological complexity $\mathrm{TC}^S$ in this class. Overall, the paper links polytope combinatorics and characteristic data to motion-planning invariants in real toric-type manifolds, providing a robust toolkit for analyzing LS-category and TC in these families.
Abstract
The real torus manifolds are a generalization of small covers and generalized real Bott manifolds. We compute the LS-category of these manifolds under some constraints and obtain sharp bounds on their topological complexities. We obtain a simplified description of their cohomology ring and discuss a relation on the cup-product of its generators. We obtain the sharp bounds on their zero-divisors-cup-lengths. We improve the dimensional upper bound on their topological complexity. We also show that under certain hypotheses, the topological complexity of real torus manifolds of dimension $n$ is either $2n$ or $2n+1$. We compute the $\mathbb{Z}_2$-equivariant LS-category of any small cover when the $\mathbb{Z}_2$-fixed points are path connected. We then compute the LS-category of Dold manifolds of real torus type and obtain sharp bounds on their topological complexity. In the end, we obtain sharp bounds on the symmetric topological complexity of a class of these manifolds.