Positive Scalar Curvature and crystallographic fundamental groups
Noe Barcenas, Mario Velasquez
TL;DR
The paper investigates the unstable Gromov-Lawson-Rosenberg conjecture for crystallographic groups formed as split extensions $1\to \mathbb{Z}^n\to \Gamma=\mathbb{Z}^n\rtimes_{\rho}\mathbb{Z}/m\to \mathbb{Z}/m\to 1$, establishing positive results in both even and odd dimensions under a positivity condition on the action when $m$ is odd, and constructing infinitely many counterexamples when $m$ is square-free. Central to the analysis is the connective KO/ko framework for $B\Gamma$, the Baum–Connes assembly map, and precise control of torsion via localized spectral sequences, which yields structural information on $ko_*(B\Gamma)$ and the finite generation of $H^{2r+1}(\underline{B}\Gamma)$. The odd- and even-dimensional positive results rely on collapsing Atiyah–Hirzebruch spectral sequences and surjectivity/injectivity arguments for the $D$-map, enabling surgery arguments to produce metrics of positive scalar curvature under the vanishing of $\alpha(M)$. Conversely, the paper extends Schick’s method and SV’s cohomology results to exhibit a broad class of counterexamples in the crystallographic setting, showing the instability of GLR in this family. These results illuminate the precise algebraic-topological obstructions controlling psc metrics for crystallographic groups and clarify the limits of GLR in this context.
Abstract
We examine positive and negative results for the Gromov-Lawson-Rosenberg Conjecture within the class of crystallographic groups. We give necessary conditions within the class of split extensions of free abelian by cyclic groups to satisfy the unstable Gromov-Lawson-Rosenberg Conjecture. We also give necessary conditions within the same class of groups, producing an infinite number of counterexamples for the conjecture.