I think I've encountered a question about 4-manifolds which maybe easy but I'm not familiar with. Can anyone give me an example of a simply connected 4-manifold $M$ (with boundary, of course) with $H\_2(M)\cong \mathbb{Z}\_k$? It must exist, I thought?


Thanks. 



Simply connected 4-manifolds with boundary
That can't happen if $k>1$. First, notice that $H\_2(M, \partial M) \cong H^2(M) \cong 0$ by Poincare duality and universal coefficients. Then the exact sequence on homology for $(M, \partial M)$ forces $H\_1(\partial M) = 0$, and then $H\_2(\partial M) = 0$, and then you have a contradiction unless $k=1$. (I'm assuming that you mean a compact manifold.)


