# Path components of space of almost complex structures

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 $$S^6$$?

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 $$w_{2}(M)$$.