Examples of 6-manifolds without an almost complex structure Question: I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure.
Finding such an example is equivelant to finding a manifold where the image of the Bockstein homomorphism $H^{2}(X,\mathbb{Z}_2) \rightarrow H^{3}(X,\mathbb{Z})$ maps the second Stiefel-Whitney class to something non-zero (this is due to Wall).
 A: Turning comments into answer: An example of a closed 6-manifold not admitting an almost complex structure is $S^1 \times (SU(3)/SO(3))$. From the obstruction theory for lifting the map $M \to BSO(6)$ classifying the tangent bundle of an oriented 6-manifold through $BU(3) \to BSO(6)$, one sees that the unique obstruction to the existence of an almost complex structure is the Bockstein of $w_2(M)$, also known as the third integral Stiefel-Whitney class $W_3(M)$. This is generally, in all dimensions, the unique obstruction for an orientable manifold to admit what is called a spin$^c$ structure.
The manifold $SU(3)/SO(3)$ does not admit a spin$^c$ structure (see e.g. Friedrich's "Dirac operators in Riemannian geometry" p.50). Also, a calculation shows that for orientable manifolds $M$ and $N$, the product $M\times N$ is spin$^c$ if and only if each factor is. Hence $S^1 \times (SU(3)/SO(3))$ is not spin$^c$, and thus not almost complex.
To create a simply connected example, one can surger out e.g. any circle of the form $S^1 \times pt$ in $S^1 \times (SU(3)/SO(3))$; note that $SU(3)/SO(3)$ is simply connected as can be seen from the homotopy long exact sequence for $SU(3)/SO(3) \to BSO(3) \to BSU(3)$. The process of taking a manifold, crossing with a circle, and then surgering out such a circle, is sometimes referred to as spinning the original manifold. Spinning a closed orientable manifold produces a spin$^c$ manifold iff the original manifold was spin$^c$, see Proposition 2.4 here https://arxiv.org/abs/1805.04751.
To create more examples for free, you can use the fact that the connected sum $M\# N$ of orientable manifolds is spin$^c$ iff each factor is.
