2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70395Let B_n be the braid group on n > 3 strands. We prove that B_n modulo its center is co-Hopfian. We then show that any injective endomorphism of B_n is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from B_n to B_n+1 is geometric. Additionally, we obtain analogous results for mapping class groups of punctured spheres. The methods use Thurston's theory of surface homeomorphisms and build upon work of Ivanov and McCarthy.27 pages, 7 figures, improved exposition, minor correctionsGeometric TopologyGroup Theory20F36; 57M07text