Let $M$ be a closed orientable connected topological manifold of dimension $4k$ with $k > 1$. It is known ([David Frank, On the index of a tangent 2-field][1]) that if $M$ has a smooth structure, the necessary and sufficient condition for it to admit two linearly independent vector fields is the following:

> The Euler characteristic $\chi(M) = 0$ and the signature $\sigma(M)
 \equiv 0 \pmod{4}$.


Do the same two conditions characterize whether the tangent microbundle $\tau M$ of $M$ has a rank 2 trivial subbundle? 

The vanishing $\chi(M) = 0$ is evidently necessary, for instance by the Lefschetz fixed point theorem; it is also sufficient for $\tau M$ to have a rank **one** trivial subbundle by Corollary 1.5 in [Ronald J. Stern, On topological and piecewise linear vector fields][2]. 

Duane Randall has papers that work out the analogous characterization with non-orientable manifolds and manifolds of dimension $4k+1$, $4k+2$, $4k+3$, but they seem to leave the orientable 4k case unsettled in general.
<br>
[Duane Randall, CAT 2-fields on nonorientable CAT manifolds][3]
<br>
[Duane Randall, On indices of tangent fields with finite singularities][4]
<br>
[Duane Randall, On 4-manifolds and span-related numbers for CAT manifolds][5]


  [1]: https://www.sciencedirect.com/science/article/pii/0040938372900110
  [2]: https://www.sciencedirect.com/science/article/pii/0040938375900075
  [3]: https://academic.oup.com/qjmath/article-abstract/38/3/355/1533968
  [4]: https://link.springer.com/chapter/10.1007/BFb0081477
  [5]: https://link.springer.com/article/10.1007/BF02567932