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Why doesn't $\mathbb CP^2 \mathbin\# \mathbb CP^2$ admit an almost complex structure?

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up vote 9 down vote accepted

According to the comments here, a necessary condition for a $4$-manifold to admit an almost complex structure is that $e+\tau = 0 \pmod 4$, where $e$ is the Euler characteristic and $\tau$ is the signature. (I don't know of a reference for that fact).

But $\mathbb{C}P^2\sharp \mathbb{C}P^2$ has euler Characteristic $4$ (its homology groups are that of $S^2\times S^2$) and its signature $\tau$ is $2$ (since the identity represents the bilinear form in the "usual" basis of $H^2(\mathbb{C}P^2\sharp\mathbb{C}P^2)$.)

Then $e+\tau = 6 \cong 2 \pmod 4$.

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