Topological defects that form on surfaces of ordered media, dubbed boojums, are ubiquitous in superfluids, liquid crystals (LCs), Langmuir monolayers, and Bose-Einstein condensates. They determine supercurrents in superfluids, impinge on electrooptical switching in polymerdispersed LCs, and mediate chemical response at nematic-isotropic fluid interfaces, but the role of surface topology in the appearance, stability, and core structure of these defects remains poorly understood. Here, we demonstrate robust generation of boojums by controlling surface topology of colloidal particles that impose tangential boundary conditions for the alignment of LC molecules. To do this, we design handlebody-shaped polymer particles with different genus g. When introduced into a nematic LC, these particles distort the nematic molecular alignment field while obeying topological constraints and induce at least 2g â 2 boojums that allow for topological charge conservation. We characterize 3D textures of boojums using polarized nonlinear optical imaging of molecular alignment and explain our findings by invoking symmetry considerations and numerical modeling of experiment-matching director fields, order parameter variations, and nontrivial handle-shaped core structure of defects. Finally, we discuss how this interplay between the topologies of colloidal surfaces and boojums may lead to controlled self-assembly of colloidal particles in nematic and paranematic hosts, which, in turn, may enable reconfigurable topological composites.eing inspired by Lewis Carroll's poem The Hunting of the Snark, Mermin (1, 2) introduced a term "boojum" to name elusive at the time surface defects in superfluids. This term and the concept of boojums quickly penetrated different fields of physics and materials science, ranging from liquid crystals (LCs) to Langmuir monolayers, and to Bose-Einstein condensates (3-6). However, unlike in the case of their bulk topological counterparts (7-10), called "hedgehog" point defects, the appearance of boojums is rarely controlled at will. In LCs, boojums can spontaneously appear at their interfaces with isotropic media, such as water. They are commonly associated with the geometries of thin LC films on surfaces of isotropic fluids (11), LC droplets (3, 4, 12), and colloidal inclusions (13,14). For example, spherical surfaces with tangential boundary conditions for rod-like LC molecules, and director n describing their local average orientation, are known to induce two boojums per LC droplet or per colloidal inclusion in the LC host (3,4,13,14). However, despite the recent progress in generating and controlling bulk LC defects using director realignment with focused laser beams, chirality, and surface topology of colloidal particles with perpendicular surface anchoring introduced into LCs (7-10), similar control of boojums has not been demonstrated.In this work, we fabricate polymer microparticles with the topology of a handlebody of genus g ranging from one to five that are capable of imposing tangentially degenerate s...