2-connected 6 manifolds with boundary $S^5$

What are the 2-connected 6-manifolds that have boundary $S^5$? Are they all of the form $(\sharp_{i=1}^k S^3 \times S^3) \backslash D^6$ for some $k \ge 1$?

Also, I think if $M^5$ is simply-connected, has $w_2(M) = 0$, and is not $S^5$, then there is a unique 2-connected 6-manifold with boundary $M$. This seems to follow from Smale's classification of simply-connected 5-manifolds, Section 6.
http://www.jstor.org/stable/pdf/1970417.pdf?acceptTC=true. But I have never heard such a claim before and it seems suspicious (especially in light of the $S^5$ case). Any references or counterexamples would be much appreciated.

• I think all 2-connected 6-manifolds that have boundary $S^5$ are indeed of the form $(\sharp_{i=1}^k S^3\times S^3) \backslash D^6$ by Smale's classification (which I was misinterpreting). Also, if $M^5$ is simply-connected and has $w_2(M) = 0$, then all 2-connected 6-manifolds with boundary $M^5$ are unique up to boundary connected sum with one of the $(\sharp_{i=1}^k S^3\times S^3) \backslash D^6$. So my original thinking was incorrect; there is not a unique 2-connected filling of $M \ne S^5$ but all such fillings are related to fillings of $S^5$. – user39598 Mar 5 '16 at 1:50

Every diffeomorphism of $S^5$ extends across $B^6$ (because there are no exotic 6-spheres), so there is a unique way of capping off the boundary. Now you're just asking what the 2-connected 6-manifolds are. Smale classified these, and they are what you expected. See here.
• the link you've reffered is not opening, I suppose you want to reffer this theorem of Smale's that if $W$ is a compact simply connected $n-$ manifold for $n\geq 6$ and with a simply connected boundary, then $W$ is diffeomorphic with $D^n$ iff $W$ has a integral homology of a point. – Anubhav Mukherjee Jun 22 '16 at 7:14
• @Anubhav The link works fine for me. The classification of 2-connected 6-manifolds is that they're diffeomorphic to one of $\#_k (S^3 \times S^3)$. – user98602 Jun 22 '16 at 7:27