Differential Topology and the Poincaré-hopf Theorem

In this paper we approach the topology of smooth manifolds using differential tools, as opposed to algebraic ones such as homology or the fundamental group. The main result is the Poincaré-Hopf index theorem, which states the sum of the indices of a vector field with finitely many zeros on a smooth compact oriented boundaryless manifold is equal to the Euler characteristic of the manifold. To lead up to this theorem, we will look at smooth maps between manifolds and study intersection numbers, fixed points, and transversality. If f : X −→ Y is a smooth map of oriented manifolds with Z a submanifold of Y , the intersection number of f with Z, denoted I(f, Z) is the number of points in f−1(Z) counted with signs ±1 depending on the way the map f behaves locally with respect to the orientations on X,Y, and Z. As a simple example, let f : S −→ R be a simple closed curve and let Z be the unit circle as a submanifold of R. The intersection of f with Z is counted positively at a point of intersection if the positive tangent vector to f and the positively oriented tangent vector to Z together form a positively oriented basis for the tangent space of R. Consequently, whenever f travels from outside the unit circle to inside the unit circle, the intersection is positive, and whenever f travels from inside the unit circle to outside the unit circle, the intersection is negative (or vice versa, depending on the chosen orientations). In order to ensure that the set f−1(Z) is finite, we will have to assume that dimX + dimZ = dimY and that f is transversal to Z. We say f is transversal to Z if for all x ∈ f−1(Z), imdfx + Tf(x)Z = Tf(x)Y , where the + denotes the span of two subspaces. Transversality intuitively means that the tangent plane to f and the tangent plane to Z at given point in imf ∩Z is not contained in any hyperplane in the ambient tangent space. We will study fixed points and Lefschetz theory as a way of proving the PoincaréHopf theorem. The proof of the Poincaré-Hopf theorem consists of two stages. First we show that the global Lefschetz number of a smooth map is equal to the sum of its local Lefschetz numbers, and provide a concrete way to compute these local Lefschetz numbers as the degrees of maps defined on local spheres. The second part of the proof involves vector fields. We will show that the degree of a vector field is equal to the global Lefschetz number of its flow. Rather than using integral curves or solutions to differential equations, we will construct a more rudimentary deformation of the of the identity that is only tangent to the vector field at time zero but that will still suffice. Since a deformation of the identity is by definition homotopic to the identity and since intersection number is homotopy