Classification of higher dimensional manifolds It is known that a 2-connected closed smooth 6-manifold is homeomorphic to S^{6} 
or connected sum of (S^{3}xS^{3}). My question is whether we have a similar statement for (n-1)-connected closed smooth 2n-manifold (at least when n=4). If not, do we have a clear classification theorem?
 A: There is a series of papers from the mid-1960's by C.T.C. Wall on classification of highly connected smooth manifolds, starting with Wall, C. T. C., Classification of (n−1)-connected 2n-manifolds. Ann. of Math. vol. 75 1962 163–189. I think this paper answers your question.
The Math Review of this paper by Kervaire starts, "This paper is an application of almost everything known in differential topology to the problem of classifying (n−1)-connected differential 2n-manifolds under diffeomorphism." The classification of 2-connected 6-manifolds to which you refer is actually a later result in this series.
Wall addresses the implications of his results to PL classification; presumably, later developments about topological manifolds would give the classification up to homeomorphism.  There are probably more `modern' ways of formulating Wall's results, as the paper was written in the early days of surgery theory.  It would be a worthwhile exercise to compare the way in which classification results are done these days with the methods of the original paper.
