Fig. 2. Illustration of no-open-edge condition and bubbling equivalence.

a One symmetric assembly in $E_{2,1}^2$ for space group Pm in one unit cell. The figure shows the cross section of a unit cell that is perpendicular to the z direction. The arrows show the orientation of 2- and 1-cells. This assembly fails the no-open-edge condition. b The evolution of assembly in kerd₁ for space group Pm in a fermionic system. The 2-cells are decorated with Chern insulators, the Chern numbers of which are indicated. This state is equivalent to an assembly in $E_{3,1}^2$, where two 3-bubbles are decorated to the 3-cells. The outward (inward) thin arrow means that the Chern number on the associated boundary is  +1 (−1).