Manifolds as open books

Introduction.We prove that all closed, simply-connected, differentiable (or p.l.) manifolds of dimension > 6 and index x = 0 decompose in a certain way: as "open books", a decomposition analogous to the classical Lefschetz decomposition of nonsingular algebraic varieties.The condition T = 0 is also necessary for a manifold to be an open book, and so, in particular, we have found a simple, intrinsic, geometric equivalent of it.For any orientable 3-manifold, this decomposition had been given by Alexander in 1923, using properties of branched coverings, which do not seem to generalize to higher dimensions.The proof of our theorem is not difficult and is a natural consequence of decomposing manifolds à la Heegaard, first accomplished for a large class of high-dimensional manifolds by Smale and completed by othersWe wish to thank W. D. Neumann, J. W. Wood and especially Professor D. Sullivan of M.I.T. for very helpful conversations.