Continuity and equivariant dimension
Alexandru Chirvasitu, Benjamin Passer
TL;DR
The paper investigates local-triviality dimensions as noncommutative analogues of Borsuk-Ulam-type invariants for actions on unital $C^*$-algebras, focusing on how these invariants behave under deformations and across continuous fields. It defines and analyzes several variants, including the local-triviality dimension $\mathrm{dim}^{{\mathbb G}}_{\mathrm{LT}}$, the weak version $\mathrm{dim}^{{\mathbb G}}_{\mathrm{WLT}}$, and the strong variant, and establishes that finiteness implies freeness while freeness does not guarantee finite weak LT dimension. Through spectral-subspace formulations for $\mathbb Z/k$-actions on matrix bundles and the study of polar-deformed objects like noncommutative tori $C({\mathbb T}^n_{\theta})$ and theta-spheres $C({\mathbb S}^{2n-1}_{\theta})$, the work demonstrates upper semicontinuity of the weak LT dimension in certain field contexts but also reveals pervasive discontinuities across deformation parameters. In the rational $\theta$-case, the paper shows striking phenomena such as $\mathrm{dim}^{\mathbb Z/q}_{\mathrm{SLT}}(C({\mathbb S}^{3}_{\theta}))=\infty$, while $\mathbb Z/2$-actions can yield finite LT/WT values (often 1) for nonintegral odd-denominator $\theta$, shedding light on the nuanced relationship between deformation, freeness, and equivariant dimension. Overall, the results advance noncommutative Borsuk-Ulam theory by clarifying how these invariants behave under deformations and by providing concrete computations for key examples like rational tori and spheres.
Abstract
We study the local-triviality dimensions of actions on $C^*$-algebras, which are invariants developed for noncommutative Borsuk-Ulam theory. While finiteness of the local-triviality dimensions is known to guarantee freeness of an action, we show that free actions need not have finite weak local-triviality dimension. Moreover, the local-triviality dimensions of a continuous field may be greater than those of its individual fibers, and the dimensions may fail to vary continuously across the fibers. However, in certain circumstances upper semicontinuity of the weak local-triviality dimension is guaranteed. We examine these results and counterexamples with a focus on noncommutative tori and noncommutative spheres, both in terms of computation and theory.