Abstract Let $M$ be a closed, odd GKM$_3$ manifold of non-negative sectional curvature. We show that in this situation one can associate an ordinary abstract graph to and prove if is orientable, then both the equivariant rational cohomology split off odd-dimensional sphere.

We explore the physics of novel fermion liquids emerging from conducting networks, where 1D metallic wires form a periodic 2D superstructure. Such structure naturally appears in marginally twisted bilayer graphenes, moire transition metal dichalcogenides, and also some charge-density wave materials. For these network systems, we theoretically show that remarkably wide variety new non-Fermi emer...