Example of a real orientable $2n$-plane bundle without complex structure via non-trivial odd Stiefel-Whitney class

Question

For any complex vector bundle, the odd Stiefel-Whitney classes of its underlying real vector bundle are trivial. So if a real vector bundle has a non-trivial odd Stiefel-Whitney class, it is not isomorphic to the underlying real vector bundle of a complex vector bundle, i.e. it has no complex structure. It is a nice result, and I would want to find a non-trivial example of a real vector bundle $\omega$ for which we can apply the result to conclude that $\omega$ has no complex structure. However, all the examples that I can think of are either non-orientable (which is equivalent to $w_1(\omega) \neq 0$), or all the odd Stiefel-Whitney classes are trivial. Hence the question reduces to this: is there a simple example of real orientable $2n$-plane bundle with a non-trivial odd Stiefel-Whitney class?

Answer 1

The universal example is the tautological bundle over $BSO(n)$. The $\mathbb{Z}/2$ cohomology of $BSO(n)$ can be calculated to be $\mathbb{Z}/2 [w_2,w_3,w_4,\dots, w_n]$. One can find a host of other examples by doing things like taking skeleta of $BSO(n)$ or taking Whitehead covers, etc.

Answer 2

Conner's answer regarding $BSO(n)$ is fine (and I voted it up), but if you wanted a closed, simply connected manifold, here's an example:

Consider the Wu manifold $M:=SU(3)/SO(3)$ (see this for more info). This is a $5$-manifold with the $\mathbb{Z}/2\mathbb{Z}$ cohomology isomorphic to that of $S^2\times S^3$. On the other hand, $w_2(TM)$ is (somewhat) famously non-zero. In addition, from the relation $Sq^1(w_2) = w_3$, together with identifying $Sq^1$ with the Bockstein tells you that $w_3(TM)\neq 0$.

Thus, $TM$ doesn't admit a complex structure. Of course, this is obvious because it's odd-dimensional, but one can, of course, consider instead $TM\oplus 1$.

If you want an example of a closed simply connected even-dimensional manifold whose tangent bundle isn't complex because of an odd-degree Stiefel-Whitney class, simply consider $M\times S^{2k+1}$ for $k\geq 1$.