A Giambelli-type formula for subbundles of the tangent bundle

Boris Shapiro and Maxim Kazarian

Vol. 230 (2007), No. 1, 233–255

DOI: 10.2140/pjm.2007.230.233

Abstract

Consider a generic n-dimensional subbundle 𝒱 of the tangent bundle TM on some manifold M. Given 𝒱, one can define different degeneracy loci Σ_r(𝒱),  r = (r₁ ≤ r₂ ≤ r₃ ≤⋯ ≤ r_k), on M consisting of all points x ∈ M for which the subspaces 𝒱^j(x) ⊂ TM(x) spanned by all length ≤ j commutators of vector fields tangent to 𝒱 at x has dimension less than or equal to r_j. Under a certain transversality assumption, we explicitly calculate the ℤ₂-cohomology classes of M dual to Σ_r(𝒱), using determinantal formulas due to W. 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.

Keywords

n-subbundles, free Lie algebra, determinantal formulas

Mathematical Subject Classification 2000

Primary: 57R20, 57R22

Milestones

Received: 13 May 2005

Revised: 10 August 2005

Accepted: 28 July 2005

Published: 1 March 2007