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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.

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  • $\begingroup$ $\mathbb Z_k$ means what ? $\mathbb Z^k$ or $\mathbb Z/k\mathbb Z$? $\endgroup$
    – DLIN
    Commented Sep 21, 2016 at 2:16
  • $\begingroup$ $\mathbb{Z}/k\mathbb{Z}$ @DLIN $\endgroup$
    – Ivy
    Commented Sep 21, 2016 at 2:19

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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.)

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