Boundary geometry controls topological defect transitions that determine 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, where bulk-surface coupling can render the geometry of confining boundaries relevant. This is particularly important in living matter, where 2D topological defects have been linked to essential biological functions, whereas the role of 3D defects is unclear. Motivated by multicellular systems interacting with extracellular boundaries, we consider a polar fluid confined within curved boundaries imposing weak surface anchoring. We report a novel charge-preserving transition between different defect configurations, controlled by the boundary shape, and invariant to changes in the material parameters. We test if this geometry-driven transition occurs in confined multicellular systems and investigate the biological role of 3D polar defects in the mouse epiblast -- an embryonic tissue consisting of apico-basally polarised cells. We find that fluid-filled lumina -- structures essential for subsequent embryonic development -- tend to form near defect positions of polar fluids in embryo-like confinement geometries. Moreover, by experimentally perturbing embryo shape beyond the transition point, we trigger the formation of additional lumen nucleation sites at the predicted position. Thus, our work reveals how boundary geometry controls polar defects, and how embryos use this mechanism for shape-dependent lumen formation. Because this defect control principle is independent of specific material properties, we expect it to apply universally to systems with orientational order.