Consider a generic n-dimensional subbundle V of the tangent bundle TM on some manifold M. Given V, one can define different degeneracy loci Σr (V), r=(r1 ≤ r2 ≤ r3 ≤· · ·≤ rk), on M consisting of all points x ε M for which the subspaces Vj (x)⊂TM (x) spanned by all length ≤ j commutators of vector fields tangent to V at x has dimension less than or equal to rj . Under a certain transversality assumption, we explicitly calculate the Z2- cohomology classes of M dual to Σr (V), using determinantal formulas due toW. Fulton and the expression of the Chern classes of the associated bundle of the free Lie algebras in terms of the Chern classes of V.
- Determinantal formulas
- Free Lie algebra