Cobordism obstructions to independent vector fields

TL;DR

This paper introduces a cobordism-based invariant that determines the existence of multiple independent vector fields on closed manifolds, linking obstruction theory with cobordism and homotopy groups.

Contribution

It defines a new invariant for vector field existence, computes it explicitly for low r, and relates it to classical cobordism and homotopy invariants.

Findings

The invariant is the top obstruction for low r cases.

Explicit computations of homotopy groups of Thom spectra are provided.

The invariant relates to well-known cobordism invariants.

Abstract

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.