On simplicial maps of the complexes of curves of nonorientable surfaces

Abstract

Let N be a compact, connected, nonorientable surface of genus g with n boundary components and C(N) be the complex of curves of N. Suppose that g + n ≤ 3 or g + n ≥ 5. If λ: C(N) → C(N) is an injective simplicial map, then λ is induced by a homeomorphism of N. If (g, n) ≠ (1,2) and λ: C(N) → C(N) is a simplicial map that satisfies the connectivity property, then λ is induced by a homeomorphism of N.