Superinjective Simplicial Maps of the Two-sided Curve Complexes on Nonorientable Surfaces

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Let N be a compact, connected, nonorientable surface of genus g with n boundary components with g ≥ 5, n ≥ 0. Let T(N) be the two-sided curve complex of N. If λ:T(N) → T(N) is a superinjective simplicial map, then there exists a homeomorphism h: N → N unique up to isotopy such that H(α) = λ(α) for every vertex α in T(N) where H=[h].