Almost complex structure on $\mathbb CP^2 \mathbin\# \mathbb CP^2$

Question

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

Answer 1

According to the comments here, a necessary 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 $e+\tau = 6 \cong 2 \pmod 4$.