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Let $M$ be a closed oriented smooth 4-manifold which admits an almost complex structure. The Ehresmann-Wu theorem states that a class $c\in H^2(M;\mathbb{Z})$ is realizable as the first Chern class of some almost complex structure $M$ if and only if $$w_2(TM)=c \mod 2 \qquad \text{and} \qquad c^2=2\chi+3\sigma, $$ where $\chi$ and $\sigma$ are the Euler characteristic and signature of $M$. Let $$ \mathcal{J}(M,c)=\{J\in \mathcal{J}(M)\mid c_1(TM,J)=c, \quad J\; \text{ is orientation compatible}\} $$ be the set of almost complex structures on $M$ with first Chern class $c$ and which are compatible with the orientation of $M$.

Then it can be shown that $$ \pi_0(\mathcal{J}(M,c))\cong \mathrm{Tor}_2(H^2(M;\mathbb{Z}))\times \left(\mathbb{Z}/2\mathbb{Z}\oplus \frac{H^3(M;\mathbb{Z})}{H^1(M;\mathbb{Z})\cup c}\right). $$ See for example section 4.1 of McDuff-Salamon's Introduction to Symplectic Topology. In particular, if $M$ is simply connected, then $\mathcal{J}(M,c)$ has precisely two connected components.

My question: Is there an analogous formula for $\pi_0(\mathcal{J}(M,c))$ for 6-manifolds (maybe with more assumptions like simply connected)? If what happens in the cases of $\mathbb{CP}^3$ or $S^6$?

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See proposition 8 of "CUBIC FORMS AND COMPLEX 3-FOLDS" by Okonek and Van-de-Ven. I restate it here for completeness.

Suppose $M$ is an oriented smooth $6$-manifold without $2$-torsion in $H^{3}(M,\mathbb{Z})$ then there is $1:1$ correspondence:

Homotopy classes of almost complex structures on $M$ $\leftrightarrow $ Integral lifts of $w_{2}(M)$.

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