Geometric residue theorems for bundle maps (Q1304452)

By geometric residue theorems we understand theorems which associate topological or curvature invariants to the singularities of geometric objects, like, for instance, the classical Hopf's theorem which relates the zeros of a vector field on a compact manifold to its Euler characteristic. In the paper under review the author studies singularities of maps between bundles over an oriented manifold obtaining residue theorems for these singularities. The paper is based on the theory of generic bundle atomic maps developed by Harvey, Lawson and Semmes where singularities have expected co-dimensions. In the present paper these results are generalized to the possibly non-generic case when singularities are allowed to be closed submanifolds possibly of different dimensions. Let \(\alpha: E\to F\) be a smooth bundle map between vector bundles with connections over a compact, oriented manifold \(X\). Suppose that \(\alpha\) is injective on \(X-\Sigma\), where \(\Sigma\) is a union of smooth closed submanifolds \(\Sigma_i\) possibly of different dimensions. The map \(\alpha\) is called normalizable at \(\Sigma\) if \(\alpha\) and connections of \(E\) and \(F\) are radially constant with respect to the natural projection \(\rho:N_\varepsilon - \Sigma\to\partial N_\varepsilon\) of \(\varepsilon\)-tubular neighborhood \(N_\varepsilon\) of \(\Sigma\). The main result of the paper is that for \(\alpha\) normalizable at each \(\Sigma_i\) and arbitrary Chern-Weil invariant polynomial \(\Phi\) there is an equation between currents on \(X\) as follows: \[ \Phi(\Omega_F) - \Phi(\Omega_E \oplus \Omega_{I^{\bot}}) = \sum \text{Res}_{\Phi,i}[\Sigma_i] + dT \] where \(I=\text{Im}(\alpha)\) is the image subbundle of \(\alpha\), \(T\) is an \(L^1_{loc}\) transgression form on \(X\), and the residue \(\text{Res}_{\Phi,i}\) is a smooth form on \(\Sigma_i\).