Representing Homology Automorphisms of Nonorientable Surfaces

In this paper, we prove that every automorphism of the first homology group of a closed, connected, nonorientable surface which preserves the associated Z2-valued intersection pairing is induced by a diffeomorphism of this surface. 0. Introduction It is well known that the diffeomorphisms on a closed, connected, orientable surface of genus g, Mg, induce the full group of automorphisms of H1(Mg,Z) which preserve the associated intersection pairing. With respect to a standard basis of H1(Mg,Z), this group is identified with the group of integer symplectic matrices, Sp(2g,Z). Clebsch and Gordon discovered generators for Sp(2g,Z) in 1866. Consequently, in 1890 Burkhardt [BU] gave the first proof of this fact by showing that these generators are induced by diffeomorphisms of Mg. A similar algebraic proof involves the set of four generators discovered by Hua and Reiner [HR], [Bi]. Meeks and Patrusky [MP] gave a topological proof in 1978. In the case of a closed, connected, nonorientable surface of genus p, Fp, there is only a Z2-valued intersection pairing. (Here, the genus of a nonorientable surface is defined to be the number of projective planes in a connected sum decomposition.) Nevertheless, we shall show in this article that the above result extends in a natural way to nonorientable surfaces. More precisely, we shall prove the following theorem. Theorem 1. If L is an automorphism of H1(Fp,Z) which preserves the associated Z2-valued intersection pairing, then L is induced by a diffeomorphism of Fp. Date: February 26, 2004. 1991 Mathematics Subject Classification. Primary 57N05; Secondary 57N65.