  We define an invariant for the existence of r pointwise linearly independent
sections in the tangent bundle of a closed manifold. For low values of r,
explicit computations of the homotopy groups of certain Thom spectra combined
with classical obstruction theory identifies this invariant as the top
obstruction to the existence of the desired sections. In particular, this shows
that the top obstruction is an invariant of the underlying manifold in these
cases, which is not true in general. The invariant is related to cobordism
theory and this gives rise to an identification of the invariant in terms of
well-known invariants. As a corollary to the computations, we can also compute
low-dimensional homotopy groups of the Thom spectra studied by Galatius,
Tillmann, Madsen, and Weiss.
