# Example of open manifold with no free integer homology non-homeomorphic to a ball

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 $$dim M >2$$.

• 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. – Anubhav Mukherjee Oct 27 '18 at 15:37
• Are there examples that are oriented? – RGC Oct 27 '18 at 16:01
• 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. – Najib Idrissi Oct 27 '18 at 16:01
• Does that kill the last homology group, Najib? – RGC Oct 27 '18 at 16:04
• @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/…. – Igor Belegradek Oct 27 '18 at 18:53