It is known that a 2connected closed smooth 6manifold is homeomorphic to S^{6} or connected sum of (S^{3}xS^{3}). My question is whether we have a similar statement for (n1)connected closed smooth 2nmanifold (at least when n=4). If not, do we have a clear classification theorem?

2$\begingroup$ I think there's statements to this effect in Kosinski's Differential Topology text, have you looked? The idea is to cancel handles of low and high dimensions, leaving only the middledimensional handles. $\endgroup$ – Ryan Budney May 6 '13 at 13:45

$\begingroup$ Ryan, I found a good ref called "A guide to the classication of manifolds" by M. Kreck. It indicates the 3connected closed 8manifold already could be quite complicated. You can google "surveys on surgery theory" and the first PDF contains this paper. $\endgroup$ – Allen May 7 '13 at 2:41
There is a series of papers from the mid1960's by C.T.C. Wall on classification of highly connected smooth manifolds, starting with Wall, C. T. C., Classification of (n−1)connected 2nmanifolds. 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 2nmanifolds under diffeomorphism." The classification of 2connected 6manifolds 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.

$\begingroup$ Thanks Danny. I have tried Wall's paper. But it seems not that clear to me. Meanwhile the paper did not give a classification when n is even, I suspect. $\endgroup$ – Allen May 6 '13 at 14:28

1$\begingroup$ I have not looked, but I suspect that Andrew Ranicki's book has a more modern approach. Still, I doubt that it will be readable for a novice. $\endgroup$ – Scott Carter May 7 '13 at 3:10

$\begingroup$ Hi,Scott. You mean his book: Algebraic and Geometric Surgery ? $\endgroup$ – Allen May 7 '13 at 9:22

1$\begingroup$ A more careful answer to the original question would be that there is not really going to be a good classification, especially when n is even, as Allen remarks. That's because the simplest homotopy invariant is the intersection form, and symmetric unimodular forms still defy classification; cf. MilnorHusemoller's book on the subject. Perhaps Allen could clarify his question, which asks about classification up to homeomorphism; this is probably realistic (or maybe easy) for fixed intersection form. But up to diffeomorphism it's a more complicated story. $\endgroup$ – Danny Ruberman May 7 '13 at 13:40

$\begingroup$ Thanks for your reminding, Danny. I always mean classification up to homeomorphism instead of diffeomorphism. $\endgroup$ – Allen May 8 '13 at 1:12