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The Poincaré-Hopf theorem relates the Euler-characteristic of a manifold to the local behavior of a generic vector-field on the manifold in a neighborhood of its zeroes. As a corollary of this, by taking the gradient, one can calculate the Euler-characteristic of a manifold from a local knowledge of a generic map to R around its singular points. We prove an analogue of this theorem for calculation of the de Rham invariant of 4k + 1 dimensional orientable manifolds from a map toi?. For 4k + 1 dimensional orientable manifolds we have the de Rham invariant d(m). This invariant is (a) the rank of the 2-torsion in H2k(M), (b) XQ(M) x2(^0 * 2 where xF(M) is the semicharacteristic of M with coefficients in F9 (C) d(M)= [WaW«-l(»0. M l = [»2**A2*(^)> flfll. where wt(M) is the fth Stiefel-Whitney class and vt is the ith Wu class of M For the equivalence of these definitions see [L-M-P]. The de Rham invariant is important in the theory of surgery; see [M] or [M-S].

A Poincaré-Hopf type theorem for the de Rham invariant