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A Giambelli-type formula for subbundles of the tangent bundle
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Consider a generic n-dimensional subbundle V of the tangent bundle TM on some given manifold M. Given V one can define different degeneracy loci r(V), r = (r1 r2 r3 · · · rk) on M consisting of all points x 2 M for which the dimension of the subspace Vj (x) TM(x) spanned by all length j commutators of vector fields tangent to V at x is less than or equal to rj . Under a certain transversality assumption we ’explicitly’ calculate the Z2-cohomology classes of M dual to r(V) using determinantal formulas due to W. Fulton and the expression for the Chern classes of the associated bundle of the free Lie algebras in terms of the Chern classes of V.
Title: A Giambelli-type formula for subbundles of the tangent bundle
Description:
Consider a generic n-dimensional subbundle V of the tangent bundle TM on some given manifold M.
Given V one can define different degeneracy loci r(V), r = (r1 r2 r3 · · · rk) on M consisting of all points x 2 M for which the dimension of the subspace Vj (x) TM(x) spanned by all length j commutators of vector fields tangent to V at x is less than or equal to rj .
Under a certain transversality assumption we ’explicitly’ calculate the Z2-cohomology classes of M dual to r(V) using determinantal formulas due to W.
Fulton and the expression for the Chern classes of the associated bundle of the free Lie algebras in terms of the Chern classes of V.
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