Let N be a connected nonorientable surface of genus g with n punctures. Suppose that g is odd and g + n 6. We prove that the automorphism group of the complex of curves of N is isomorphic to the mapping class group MN of N .
Let Ng be the connected closed nonorientable surface of genus g ≥ 5 and Mod(Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod(Ng) is either trivial or Z if g is odd, and injects into the mapping class group of sphere with four holes if g is even.
Let S be a nonorientable surface of genus g ≥ 5 with n ≥ 0 punctures, and Mod(S) its mapping class group. We define the complexity of S to be the maximum rank of a free abelian subgroup of Mod(S). Suppose that S 1 and S 2 are two such surfaces of the same complexity. We prove that every isomorphism Mod(S 1 ) → Mod(S 2 ) is induced by a diffeomorphism S 1 → S 2 . This is an analogue of Ivanov's theorem on automorphisms of the mapping class groups of an orientable surface, and also an extension and improvement of the first author's previous result.
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