A Poincaré-Hopf theorem for noncompact manifolds

We provide the natural extension, from the dynamical point of view, of the Poincaré-Hopf theorem to noncompact manifolds. On the other hand, given a compact set K being an attractor for a flow generated by a C1 tangent vector field X on an n-manifold, we prove that the Euler characteristic of its region of attraction A, χ(A), is defined and satisfies IndA(X) = (−1)nχ(A). Finally we prove that χ(A) = χ(K) when K is an euclidean neighbourhood retract being asymptotically stable and invariant.