Generalized Poincare's Conjecture in Dimensions Greater Than Four

Poincare has posed the problem as to whether every simply connected closed 3-manifold (triangulated) is homeomorphic to the 3-sphere, see [18] for example. This problem, still open, is usually called Poincare's conjecture. The generalized Poincare conjecture (see [11] or [28] for example) says that every closed n-manifold which has the homotopy type of the nsphere S is homeomorphic to the n-sphere. One object of this paper is to prove that this is indeed the case if n > 5 (for differentiable manifolds in the following theorem and combinatorial manifolds in Theorem B).