The Index of a Vector Field as an Invariant

The purpose of this paper is to demonstrate that the index of a vector field on a smooth manifold is an intrinsic invariant of the manifold. The paper is divided into three sections. The first section introduces the Inverse Function Theorem and provides the proof for the Implicit Function Theorem, laying the foundation for our discussion. In the second section we will introduce the concept of a vector field, degree, and the index of a vector field. We will also prove that the index of a vector field is invariant under diffeomorphism. In the last section we will prove and highlight the fact that the index of a vector field on a manifold is independent of the choice of vector field or the embedding of the manifold, demonstrating that the vector field index is an invariant of a smooth manifold. The primary source for this paper was John W. Milnor’s book, Topology from the Differentiable Viewpoint.