I would like to state that if an open oriented even-dimensional (complex) manifold $M$ is such that $dim(H_k(M,\mathbb{Z}))=0$ for $k>0$, and 1 for $k=0$, then $M$ is homeomorphic to an open ball.

Is this true? Any ideas to work with? I am specially interested when $dim M >2$.

  • 1
    $\begingroup$ Freedman proved that every homology sphere bounfs a contractible 4 topological manifolds. Consider such a smooth one and take it's boundary out. Now the resultant manifold is not open ball. $\endgroup$ – Anubhav Mukherjee Oct 27 '18 at 15:37
  • $\begingroup$ Are there examples that are oriented? $\endgroup$ – RGC Oct 27 '18 at 16:01
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    $\begingroup$ Or simply remove a point from the Poincaré sphere, no? The result has nontrivial $\pi_1$ so it's certainly not homeomorphic to an open ball. $\endgroup$ – Najib Idrissi Oct 27 '18 at 16:01
  • $\begingroup$ Does that kill the last homology group, Najib? $\endgroup$ – RGC Oct 27 '18 at 16:04
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    $\begingroup$ @MikeMiller: it is actually Stallings. The reference is in en.wikipedia.org/wiki/Simply_connected_at_infinity. Siebenman of course also worked on this topic starting from his thesis, see citeseerx.ist.psu.edu/viewdoc/…. $\endgroup$ – Igor Belegradek Oct 27 '18 at 18:53

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