  Let 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.
