Cobordism Obstructions to Independent Vector Fields

boediyono;

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 the low-dimensional homotopy groups of the embedded cobordism category introduced in Galatius et al. [The homotopy type of the cobordism category, Acta Math. 202 (2009), 195–239].