Boundary geometry controls a topological defect transition that determines lumen nucleation in embryonic development

Topological defects determine the collective properties of anisotropic materials. How their configurations are controlled is not well understood however, especially in 3D. In living matter moreover, 2D defects have been linked to biological functions, but the role of 3D polar defects is unclear. Combining computational and experimental approaches, we investigate how confinement geometry controls surface-aligned polar fluids, and what biological role 3D polar defects play in tissues interacting with extracellular boundaries. We discover a charge-preserving transition between 3D defect configurations driven by boundary geometry and independent of material parameters, and show that defect positions predict the locations where fluid-filled lumina -- structures essential for development -- form within the confined polar tissue of the mouse embryo. Experimentally perturbing embryo shape beyond the transition point, we moreover create additional lumina at predicted defect locations. Our work reveals how boundary geometry controls polar defects, and how embryos use this mechanism for shape-dependent lumen formation. We expect this defect control principle to apply broadly to systems with orientational order.