Superinjective Simplicial Maps of the Complexes of Curves on Nonorientable Surfaces

We prove that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface, if $(g,n)\in \{(1,0),(1,1),(2,0),(2,1),(3,0)\}$ or $g + n \geq 5$,where$g$isthegenusofthesurfaceand$n$ is the number of the boundary components.