Abstract
If G is a compact Lie group acting smoothly on a manifold M, we prove that a G-invariant neighbourhood of the singular set Σ in M is completely determined by the G-vector bundle restriction of the tangent bundle of M to Σ. Moreover, by using only this G-vector bundle we define a residual linear map, above certain degrees, giving the Chern-Weil homomorphism for M, after composing it in cohomology with the inclusion of M in ( M, M — Σ). In the case of G being a torus, we characterize those G-vector bundles appearing as restriction to the singular set of the tangent bundle of some smooth G-manifold.
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