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We consider an oriented surface S and a cellular complex X of curves on S , defined by Hatcher and Thurston in 1980. We prove by elementary means, without Cerf theory, that the complex X is connected and simply connected. From this we derive an explicit simple presentation of the mapping class group of S , following the ideas of Hatcher-Thurston and Harer. AMS Classification numbers Primary: 20F05,20F34,57M05Secondary: 20F38,57M60

In this paper we show that even in the case of simply connected minimal algebraic surfaces of general type, deformation and differentiable equivalence do not coincide.Exhibiting several simple families of surfaces which are not deformation equivalent, and proving their diffeomorphism, we give a counterexample to a weaker form of the speculation DEF = DIFF of R. Friedman and J. Morgan, i.e., in the case where (by M. Freedman's theorem) the topological type is completely determined by the numerical invariants of the surface.We hope that the methods of proof may turn out to be quite useful to show diffeomorphism and indeed symplectic equivalence for many important classes of algebraic surfaces and symplectic 4-manifolds.

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