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Is there a non-trivial connex compact orientable topological manifold of Euler characteristic $\chi = 1$?

Remark: the point has $\chi = 1$, but it is trivial. The real projective plane has $\chi = 1$, but it is not orientable. The wedge of a sphere and a torus has $\chi = 1$, but it is not a topological manifold. I don't know if the connexity is necessary.

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$\Bbb CP^2$ has Euler characteristic $3$. Now try to use that $\chi(M \# N) = \chi(M) + \chi(N) - 2$ for even-dimensional manifolds to construct a manifold with $\chi=1$.

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    $\begingroup$ argh, just wrote mine:) anyway, clearly nice answer! $\endgroup$ – Riccardo Jun 9 '16 at 17:09
  • $\begingroup$ $M \# N$ is the connected sum. $\endgroup$ – Sebastien Palcoux Jun 9 '16 at 18:38
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$\mathbb{C}P^2\sharp S^1\times S^1 \times S^1 \times S^1 $

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